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    <title>Pinboard (cshalizi)</title>
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    <description>recent bookmarks from cshalizi</description>
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	<rdf:li rdf:resource="https://arxiv.org/abs/2101.06936"/>
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  </channel><item rdf:about="https://faculty.washington.edu/yenchic/short_note/note_MoM.pdf">
    <title>A short note on the median-of-means estimator (Yen-Chi Chen, 2020)</title>
    <dc:date>2026-04-23T16:43:42+00:00</dc:date>
    <link>https://faculty.washington.edu/yenchic/short_note/note_MoM.pdf</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA[Very nice.]]></description>
<dc:subject>to:NB have_read statistics heavy_tails estimation empirical_processes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:ee5135168ce3/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2503.21576">
    <title>[2503.21576] Empirical Measures and Strong Laws of Large Numbers in Categorical Probability</title>
    <dc:date>2025-03-31T13:50:16+00:00</dc:date>
    <link>https://arxiv.org/abs/2503.21576</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["The Glivenko-Cantelli theorem is a uniform version of the strong law of large numbers. It states that for every IID sequence of random variables, the empirical measure converges to the underlying distribution (in the sense of uniform convergence of the CDF). In this work, we provide tools to study such limits of empirical measures in categorical probability.
"We propose two axioms, permutation invariance and empirical adequacy, that a morphism of type Xℕ→X should satisfy to be interpretable as taking an infinite sequence as input and producing a sample from its empirical measure as output. Since not all sequences have a well-defined empirical measure, ``such empirical sampling morphisms'' live in quasi-Markov categories, which, unlike Markov categories, allow partial morphisms. Given an empirical sampling morphism and a few other properties, we prove representability as well as abstract versions of the de Finetti theorem, the Glivenko-Cantelli theorem and the strong law of large numbers.
"We provide several concrete constructions of empirical sampling morphisms as partially defined Markov kernels on standard Borel spaces. Instantiating our abstract results then recovers the standard Glivenko-Cantelli theorem and the strong law of large numbers for random variables with finite first moment. Our work thus provides a joint proof of these two theorems in conjunction with the de Finetti theorem from first principles."]]></description>
<dc:subject>to:NB probability empirical_processes abstract_nonsense</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:9cec99f9353f/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2501.13813">
    <title>[2501.13813] Regularizing random points by deleting a few</title>
    <dc:date>2025-02-03T00:30:24+00:00</dc:date>
    <link>https://arxiv.org/abs/2501.13813</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["It is well understood that if one is given a set $X \subset [0,1]$ of $n$ independent uniformly distributed random variables, then
$\sup_{0≤x≤1}{\left∣\frac{∣#{X \cap [0,x]}{#X}−x\right∣} \simleq \frac{\sqrt{\log{n}}{\sqrt{n}}$ with very high probability.
We show that one can improve the error term by removing a few of the points. For any $m\leq 0.001n$ there exists a subset $Y \subset X$ obtained by deleting at most $m$ points, so that the error term drops from $\sim \frac{\sqrt{\log{n}}}{n}$ to $\log(n)/m$ with high probability. When $m=cn$ for a small $0\leq c \leq 0.001$, this achieves the essentially optimal asymptotic order of discrepancy $\log{(n)}/n$. The proof is constructive and works in an online setting (where one is given the points sequentially, one at a time, and has to decide whether to keep or discard it). A change of variables shows the same result for any random variables on the real line with absolutely continuous density."]]></description>
<dc:subject>to:NB probability empirical_processes monte_carlo via:?</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:97e6f84d0342/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2403.16651">
    <title>[2403.16651] A short proof of the Dvoretzky--Kiefer--Wolfowitz--Massart inequality</title>
    <dc:date>2024-12-11T15:52:43+00:00</dc:date>
    <link>https://arxiv.org/abs/2403.16651</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["The Dvoretzky--Kiefer--Wolfowitz--Massart inequality gives a sub-Gaussian tail bound on the supremum norm distance between the empirical distribution function of a random sample and its population counterpart. We provide a short proof of a result that improves the existing bound in two respects. First, our one-sided bound holds without any restrictions on the failure probability, thereby verifying a conjecture of Birnbaum and McCarty (1958). Second, it is local in the sense that it holds uniformly over sub-intervals of the real line with an error rate that adapts to the behaviour of the population distribution function on the interval."]]></description>
<dc:subject>to:NB empirical_processes to_read re:almost_none to_teach:childs_garden_of_statistical_learning_theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:847d245f036a/</dc:identifier>
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<item rdf:about="https://link.springer.com/book/10.1007/978-3-642-03942-3">
    <title>Geometric Discrepancy: An Illustrated Guide | SpringerLink</title>
    <dc:date>2024-07-17T18:50:19+00:00</dc:date>
    <link>https://link.springer.com/book/10.1007/978-3-642-03942-3</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Discrepancy theory is also called the theory of irregularities of distribution. Here are some typical questions: What is the "most uniform" way of dis­tributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? For a precise formulation of these questions, we must quantify the irregularity of a given distribution, and discrepancy is a numerical parameter of a point set serving this purpose. Such questions were first tackled in the thirties, with a motivation com­ing from number theory. A more or less satisfactory solution of the basic discrepancy problem in the plane was completed in the late sixties, and the analogous higher-dimensional problem is far from solved even today. In the meantime, discrepancy theory blossomed into a field of remarkable breadth and diversity. There are subfields closely connected to the original number­ theoretic roots of discrepancy theory, areas related to Ramsey theory and to hypergraphs, and also results supporting eminently practical methods and algorithms for numerical integration and similar tasks. The applications in­clude financial calculations, computer graphics, and computational physics, just to name a few. This book is an introductory textbook on discrepancy theory. It should be accessible to early graduate students of mathematics or theoretical computer science. At the same time, about half of the book consists of material that up until now was only available in original research papers or in various surveys."]]></description>
<dc:subject>books:noted mathematics geometry empirical_processes via:vaguery in_NB downloaded</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:1be9139b4e2a/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2309.17016">
    <title>[2309.17016] Efficient Agnostic Learning with Average Smoothness</title>
    <dc:date>2023-12-08T17:15:57+00:00</dc:date>
    <link>https://arxiv.org/abs/2309.17016</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We study distribution-free nonparametric regression following a notion of average smoothness initiated by Ashlagi et al. (2021), which measures the "effective" smoothness of a function with respect to an arbitrary unknown underlying distribution. While the recent work of Hanneke et al. (2023) established tight uniform convergence bounds for average-smooth functions in the realizable case and provided a computationally efficient realizable learning algorithm, both of these results currently lack analogs in the general agnostic (i.e. noisy) case.
"In this work, we fully close these gaps. First, we provide a distribution-free uniform convergence bound for average-smoothness classes in the agnostic setting. Second, we match the derived sample complexity with a computationally efficient agnostic learning algorithm. Our results, which are stated in terms of the intrinsic geometry of the data and hold over any totally bounded metric space, show that the guarantees recently obtained for realizable learning of average-smooth functions transfer to the agnostic setting. At the heart of our proof, we establish the uniform convergence rate of a function class in terms of its bracketing entropy, which may be of independent interest."]]></description>
<dc:subject>in_NB nonparametrics learning_theory empirical_processes kith_and_kin kontorovich.aryeh hanneke.steve</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:372bb023a730/</dc:identifier>
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<item rdf:about="https://notstatschat.rbind.io/2022/09/28/uniform-law-of-large-numbers/">
    <title>A plug-in uniform law of large numbers - Biased and Inefficient</title>
    <dc:date>2023-06-15T19:18:32+00:00</dc:date>
    <link>https://notstatschat.rbind.io/2022/09/28/uniform-law-of-large-numbers/</link>
    <dc:creator>cshalizi</dc:creator><dc:subject>have_read ergodic_theory empirical_processes lumley.thomas learning_theory re:HEAS in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:1dfcf0d1d3f9/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2305.02960">
    <title>[2305.02960] Majorizing Measures, Codes, and Information</title>
    <dc:date>2023-05-06T22:44:46+00:00</dc:date>
    <link>https://arxiv.org/abs/2305.02960</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["The majorizing measure theorem of Fernique and Talagrand is a fundamental result in the theory of random processes. It relates the boundedness of random processes indexed by elements of a metric space to complexity measures arising from certain multiscale combinatorial structures, such as packing and covering trees. This paper builds on the ideas first outlined in a little-noticed preprint of Andreas Maurer to present an information-theoretic perspective on the majorizing measure theorem, according to which the boundedness of random processes is phrased in terms of the existence of efficient variable-length codes for the elements of the indexing metric space."]]></description>
<dc:subject>to_read empirical_processes information_theory raginsky.maxim in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:1d743b81e124/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2202.04415">
    <title>[2202.04415] Towards Empirical Process Theory for Vector-Valued Functions: Metric Entropy of Smooth Function Classes</title>
    <dc:date>2022-06-13T17:00:24+00:00</dc:date>
    <link>https://arxiv.org/abs/2202.04415</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This paper provides some first steps in developing empirical process theory for functions taking values in a vector space. Our main results provide bounds on the entropy of classes of smooth functions taking values in a Hilbert space, by leveraging theory from differential calculus of vector-valued functions and fractal dimension theory of metric spaces. We demonstrate how these entropy bounds can be used to show the uniform law of large numbers and asymptotic equicontinuity of the function classes, and also apply it to statistical learning theory in which the output space is a Hilbert space. We conclude with a discussion on the extension of Rademacher complexities to vector-valued function classes."]]></description>
<dc:subject>to:NB learning_theory empirical_processes hilbert_space via:?</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:fe711e83cc23/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2011.00308">
    <title>[2011.00308] Mixing it up: A general framework for Markovian statistics</title>
    <dc:date>2021-06-25T14:55:04+00:00</dc:date>
    <link>https://arxiv.org/abs/2011.00308</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Up to now, the nonparametric analysis of multidimensional continuous-time Markov processes has focussed strongly on specific model choices, mostly related to symmetry of the semigroup. While this approach allows to study the performance of estimators for the characteristics of the process in the minimax sense, it restricts the applicability of results to a rather constrained set of stochastic processes and in particular hardly allows incorporating jump structures. As a consequence, for many models of applied and theoretical interest, no statement can be made about the robustness of typical statistical procedures beyond the beautiful, but limited framework available in the literature. To close this gap, we identify β-mixing of the process and heat kernel bounds on the transition density as a suitable combination to obtain sup-norm and L2 kernel invariant density estimation rates matching the case of reversible multidimenisonal diffusion processes and outperforming density estimation based on discrete i.i.d. or weakly dependent data. Moreover, we demonstrate how up to log-terms, optimal sup-norm adaptive invariant density estimation can be achieved within our general framework based on tight uniform moment bounds and deviation inequalities for empirical processes associated to additive functionals of Markov processes. The underlying assumptions are verifiable with classical tools from stability theory of continuous time Markov processes and PDE techniques, which opens the door to evaluate statistical performance for a vast amount of Markov models. We highlight this point by showing how multidimensional jump SDEs with Lévy driven jump part under different coefficient assumptions can be seamlessly integrated into our framework, thus establishing novel adaptive sup-norm estimation rates for this class of processes."]]></description>
<dc:subject>to:NB to_read markov_models minimax empirical_processes statistical_inference_for_stochastic_processes re:almost_none mixing statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:60fea291f30b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:markov_models"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:minimax"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistical_inference_for_stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:almost_none"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:mixing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://projecteuclid.org/journals/annals-of-statistics/volume-49/issue-2/Empirical-process-results-for-exchangeable-arrays/10.1214/20-AOS1981.short">
    <title>Empirical process results for exchangeable arrays</title>
    <dc:date>2021-04-14T15:55:48+00:00</dc:date>
    <link>https://projecteuclid.org/journals/annals-of-statistics/volume-49/issue-2/Empirical-process-results-for-exchangeable-arrays/10.1214/20-AOS1981.short</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Exchangeable arrays are natural tools to model common forms of dependence between units of a sample. Jointly exchangeable arrays are well suited to dyadic data, where observed random variables are indexed by two units from the same population. Examples include trade flows between countries or relationships in a network. Separately exchangeable arrays are well suited to multiway clustering, where units sharing the same cluster (e.g., geographical areas or sectors of activity when considering individual wages) may be dependent in an unrestricted way. We prove uniform laws of large numbers and central limit theorems for such exchangeable arrays. We obtain these results under the same moment restrictions and conditions on the class of functions as those typically assumed with i.i.d. data. We also show the convergence of bootstrap processes adapted to such arrays."]]></description>
<dc:subject>to:NB graphons exchangeability empirical_processes bootstrap re:network_bootstraps to_read</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:704083f7b44d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:graphons"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:exchangeability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:bootstrap"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:network_bootstraps"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2101.06936">
    <title>[2101.06936] Wasserstein Convergence Rate for Empirical Measures of Markov Chains</title>
    <dc:date>2021-01-19T18:33:59+00:00</dc:date>
    <link>https://arxiv.org/abs/2101.06936</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We consider a Markov chain on ℝd with invariant measure μ. We are interested in the rate of convergence of the empirical measures towards the invariant measure with respect to the 1-Wasserstein distance. The main result of this article is a new upper bound for the expected Wasserstein distance, which is proved by combining the Kantorovich dual formula with a Fourier expansion. In addition, we show how concentration inequalities around the mean can be obtained."]]></description>
<dc:subject>to:NB markov_models empirical_processes concentration_of_measure stochastic_processes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:f9a9a23f65e4/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:markov_models"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:concentration_of_measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2012.15678">
    <title>[2012.15678] On Gaussian Approximation for M-Estimator</title>
    <dc:date>2021-01-03T20:05:42+00:00</dc:date>
    <link>https://arxiv.org/abs/2012.15678</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This study develops a non-asymptotic Gaussian approximation theory for distributions of M-estimators, which are defined as maximizers of empirical criterion functions. In existing mathematical statistics literature, numerous studies have focused on approximating the distributions of the M-estimators for statistical inference. In contrast to the existing approaches, which mainly focus on limiting behaviors, this study employs a non-asymptotic approach, establishes abstract Gaussian approximation results for maximizers of empirical criteria, and proposes a Gaussian multiplier bootstrap approximation method. Our developments can be considered as an extension of the seminal works (Chernozhukov, Chetverikov and Kato (2013, 2014, 2015)) on the approximation theory for distributions of suprema of empirical processes toward their maximizers. Through this work, we shed new lights on the statistical theory of M-estimators. Our theory covers not only regular estimators, such as the least absolute deviations, but also some non-regular cases where it is difficult to derive or to approximate numerically the limiting distributions such as non-Donsker classes and cube root estimators."]]></description>
<dc:subject>to:NB central_limit_theorem estimation statistics empirical_processes re:HEAS</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:580f8c5bf237/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:central_limit_theorem"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:estimation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:HEAS"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2012.10320">
    <title>[2012.10320] Local Dvoretzky-Kiefer-Wolfowitz confidence bands</title>
    <dc:date>2020-12-21T04:40:03+00:00</dc:date>
    <link>https://arxiv.org/abs/2012.10320</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["In this paper, we revisit the concentration inequalities for the supremum of the cumulative distribution function (CDF) of a real-valued continuous distribution as established by Dvoretzky, Kiefer, Wolfowitz and revisited later by Massart in in two seminal papers. We focus on the concentration of the \textit{local} supremum over a sub-interval, rather than on the full domain. That is, denoting U the CDF of the uniform distribution over [0,1] and Un its empirical version built from n samples, we study $\Pr\Big(\sup_{u\in [\uu,\ou]} U_n(u)-U(u) > \epsilon\Big)$ for different values of $\uu,\ou\in[0,1]$. Such local controls naturally appear for instance when studying estimation error of spectral risk-measures (such as the conditional value at risk), where $[\uu,\ou]$ is typically [0,α] or [1−α,1] for a risk level α, after reshaping the CDF F of the considered distribution into U by the general inverse transform F−1. Extending a proof technique from Smirnov, we provide exact expressions of the local quantities $\Pr\Big(\sup_{u\in [\uu,\ou]} U_n(u)-U(u) > \epsilon\Big)$ and $\Pr\Big(\sup_{u\in [\uu,\ou]} U(u)-U_n(u) > \epsilon\Big)$ for each $n,\epsilon,\uu,\ou$. Interestingly these quantities, seen as a function of ϵ, can be easily inverted numerically into functions of the probability level δ. Although not explicit, they can be computed and tabulated. We plot such expressions and compare them to the classical bound ln(1/δ)2n‾‾‾‾‾√ provided by Massart inequality. Last, we extend the local concentration results holding individually for each n to time-uniform concentration inequalities holding simultaneously for all n, revisiting a reflection inequality by James, which is of independent interest for the study of sequential decision making strategies."]]></description>
<dc:subject>to:NB empirical_processes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:c7d8f7494fc8/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2003.13530">
    <title>[2003.13530] Bounding the expectation of the supremum of empirical processes indexed by Hölder classes</title>
    <dc:date>2020-12-18T10:31:59+00:00</dc:date>
    <link>https://arxiv.org/abs/2003.13530</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["In this note, we provide upper bounds on the expectation of the supremum of empirical processes indexed by Hölder classes of any smoothness and for any distribution supported on a bounded set in ℝd. These results can be alternatively seen as non-asymptotic risk bounds, when the unknown distribution is estimated by its empirical counterpart, based on n independent observations, and the error of estimation is quantified by the integral probability metrics (IPM). In particular, the IPM indexed by a Hölder class is considered and the corresponding rates are derived. These results interpolate between the two well-known extreme cases: the rate n−1/d corresponding to the Wassertein-1 distance (the least smooth case) and the fast rate n−1/2 corresponding to very smooth functions (for instance, functions from an RKHS defined by a bounded kernel)."]]></description>
<dc:subject>to:NB empirical_processes learning_theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:38793d7ed1b5/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://projecteuclid.org/euclid.bj/1574758845">
    <title>Lei : Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces</title>
    <dc:date>2019-12-03T19:47:19+00:00</dc:date>
    <link>https://projecteuclid.org/euclid.bj/1574758845</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence on the dimensionality. Our method also covers the important case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper bounds for functional data distributions whose coordinates decay geometrically or polynomially. Moreover, our bounds of the expected value can be combined with mean-concentration results to yield improved exponential tail probability bounds for the Wasserstein error of empirical measures under Bernstein-type or log Sobolev-type conditions."]]></description>
<dc:subject>concentration_of_measure empirical_processes statistics in_NB kith_and_kin lei.jing</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:7156f6d26814/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:concentration_of_measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kith_and_kin"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:lei.jing"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1910.09319">
    <title>[1910.09319] Empirical Process of Multivariate Gaussian under General Dependence</title>
    <dc:date>2019-10-29T14:25:00+00:00</dc:date>
    <link>https://arxiv.org/abs/1910.09319</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This paper explores certain kinds of empirical process with respect to the components of multivariate Gaussian. We put forward some finite sample bounds which hold for multivariate Gaussian under general dependence. As a direct corollary, we prove that the empirical distribution of a Gaussian process will converge, that is to say,
supt|Fˆn(t)−EFˆn(t)|−→P0,
as long as the covariance of the Gaussian process vanishes with the time shift."]]></description>
<dc:subject>gaussian_processes stochastic_processes empirical_processes re:almost_none in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:ac75e1d721b1/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:gaussian_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:almost_none"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1910.07485">
    <title>[1910.07485] Excess risk bounds in robust empirical risk minimization</title>
    <dc:date>2019-10-17T14:07:49+00:00</dc:date>
    <link>https://arxiv.org/abs/1910.07485</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This paper investigates robust versions of the general empirical risk minimization algorithm, one of the core techniques underlying modern statistical methods. Success of the empirical risk minimization is based on the fact that for a "well-behaved" stochastic process {f(X), f∈} indexed by a class of functions f∈, averages 1N∑Nj=1f(Xj) evaluated over a sample X1,…,XN of i.i.d. copies of X provide good approximation to the expectations 𝔼f(X) uniformly over large classes f∈. However, this might no longer be true if the marginal distributions of the process are heavy-tailed or if the sample contains outliers. We propose a version of empirical risk minimization based on the idea of replacing sample averages by robust proxies of the expectation, and obtain high-confidence bounds for the excess risk of resulting estimators. In particular, we show that the excess risk of robust estimators can converge to 0 at fast rates with respect to the sample size. We discuss implications of the main results to the linear and logistic regression problems, and evaluate the numerical performance of proposed methods on simulated and real data."]]></description>
<dc:subject>to:NB learning_theory empirical_processes probability statistics heavy_tails to_teach:childs_garden_of_statistical_learning_theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:bbde8fa32918/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:probability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:heavy_tails"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:childs_garden_of_statistical_learning_theory"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://projecteuclid.org/euclid.bj/1568362049">
    <title>Dolera , Regazzini : Uniform rates of the Glivenko–Cantelli convergence and their use in approximating Bayesian inferences</title>
    <dc:date>2019-09-13T13:02:42+00:00</dc:date>
    <link>https://projecteuclid.org/euclid.bj/1568362049</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This paper deals with suitable quantifications in approximating a probability measure by an “empirical” random probability measure 𝔭̂ np^n, depending on the first nn terms of a sequence {ξ̃ i}i≥1{ξ~i}i≥1 of random elements. Section 2 studies the range of oscillation near zero of the Wasserstein distance d(p)[𝕊]d[S](p) between 𝔭0p0 and 𝔭̂ np^n, assuming the ξ̃ iξ~i’s i.i.d. from 𝔭0p0. In Theorem 2.1 𝔭0p0 can be fixed in the space of all probability measures on (ℝd,ℬ(ℝd))(Rd,B(Rd)) and 𝔭̂ np^n coincides with the empirical measure 𝔢̃ n:=1n∑ni=1δξ̃ ie~n:=1n∑i=1nδξ~i. In Theorem 2.2 (Theorem 2.3, respectively), 𝔭0p0 is a dd-dimensional Gaussian distribution (an element of a distinguished statistical exponential family, respectively) and 𝔭̂ np^n is another dd-dimensional Gaussian distribution with estimated mean and covariance matrix (another element of the same family with an estimated parameter, respectively). These new results improve on allied recent works by providing also uniform bounds with respect to nn, meaning the finiteness of the pp-moment of supn≥1bnd(p)[𝕊](𝔭0,𝔭̂ n)supn≥1⁡bnd[S](p)(p0,p^n) is proved for some diverging sequence bnbn of positive numbers. In Section 3, assuming the ξ̃ iξ~i’s exchangeable, one studies the range of oscillation near zero of the Wasserstein distance between the conditional distribution – also called posterior – of the directing measure of the sequence, given ξ̃ 1,…,ξ̃ nξ~1,…,ξ~n, and the point mass at 𝔭̂ np^n. Similarly, a bound for the approximation of predictive distributions is given. Finally, Theorems from 3.3 to 3.5 reconsider Theorems from 2.1 to 2.3, respectively, according to a Bayesian perspective."]]></description>
<dc:subject>empirical_processes statistics re:fitness_sampling in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:f773f86d023c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:fitness_sampling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1702.08109">
    <title>[1702.08109] Variational Analysis of Constrained M-Estimators</title>
    <dc:date>2019-09-12T16:24:45+00:00</dc:date>
    <link>https://arxiv.org/abs/1702.08109</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We propose a unified framework for establishing existence of nonparametric M-estimators, computing the corresponding estimates, and proving their strong consistency when the class of functions is exceptionally rich. In particular, the framework addresses situations where the class of functions is complex involving information and assumptions about shape, pointwise bounds, location of modes, height at modes, location of level-sets, values of moments, size of subgradients, continuity, distance to a "prior" function, multivariate total positivity, and any combination of the above. The class might be engineered to perform well in a specific setting even in the presence of little data. The framework views the class of functions as a subset of a particular metric space of upper semicontinuous functions under the Attouch-Wets distance. In addition to allowing a systematic treatment of numerous M-estimators, the framework yields consistency of plug-in estimators of modes of densities, maximizers of regression functions, level-sets of classifiers, and related quantities, and also enables computation by means of approximating parametric classes. We establish consistency through a one-sided law of large numbers, here extended to sieves, that relaxes assumptions of uniform laws, while ensuring global approximations even under model misspecification."]]></description>
<dc:subject>to:NB estimation empirical_processes statistics nonparametrics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:e5d16319e505/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:estimation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:nonparametrics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1809.06522">
    <title>[1809.06522] Concentration Inequalities for the Empirical Distribution</title>
    <dc:date>2019-09-12T00:15:30+00:00</dc:date>
    <link>https://arxiv.org/abs/1809.06522</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We study concentration inequalities for the Kullback--Leibler (KL) divergence between the empirical distribution and the true distribution. Applying a recursion technique, we improve over the method of types bound uniformly in all regimes of sample size n and alphabet size k, and the improvement becomes more significant when k is large. We discuss the applications of our results in obtaining tighter concentration inequalities for L1 deviations of the empirical distribution from the true distribution, and the difference between concentration around the expectation or zero. We also obtain asymptotically tight bounds on the variance of the KL divergence between the empirical and true distribution, and demonstrate their quantitatively different behaviors between small and large sample sizes compared to the alphabet size."]]></description>
<dc:subject>concentration_of_measure deviation_inequalities empirical_processes statistics probability to_read in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:c03a05b6c59d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:concentration_of_measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:deviation_inequalities"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:probability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1909.02088">
    <title>[1909.02088] On Least Squares Estimation under Heteroscedastic and Heavy-Tailed Errors</title>
    <dc:date>2019-09-09T03:46:50+00:00</dc:date>
    <link>https://arxiv.org/abs/1909.02088</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We consider least squares estimation in a general nonparametric regression model. The rate of convergence of the least squares estimator (LSE) for the unknown regression function is well studied when the errors are sub-Gaussian. We find upper bounds on the rates of convergence of the LSE when the errors have uniformly bounded conditional variance and have only finitely many moments. We show that the interplay between the moment assumptions on the error, the metric entropy of the class of functions involved, and the "local" structure of the function class around the truth drives the rate of convergence of the LSE. We find sufficient conditions on the errors under which the rate of the LSE matches the rate of the LSE under sub-Gaussian error. Our results are finite sample and allow for heteroscedastic and heavy-tailed errors."]]></description>
<dc:subject>to:NB regression empirical_processes statistics heavy_tails nonparametrics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:fffcfe887104/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:regression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:heavy_tails"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:nonparametrics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://link.springer.com/article/10.1007/s11203-018-9194-8">
    <title>Testing nonstationary and absolutely regular nonlinear time series models | SpringerLink</title>
    <dc:date>2019-08-28T15:10:22+00:00</dc:date>
    <link>https://link.springer.com/article/10.1007/s11203-018-9194-8</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We study some general methods for testing the goodness-of-fit of a general nonstationary and absolutely regular nonlinear time series model. These testing methods are based on some marked empirical processes that we show to converge in distribution to a zero-mean Gaussian process with respect to the Skorohod topology. We investigate the behavior of this process under fixed alternatives and under a sequence of local alternatives. Our results are applied to testing a general class of nonlinear semiparametric time series models. A simulation experiment shows that the Cramér–von Mises test studied behaves well on the examples considered."]]></description>
<dc:subject>to:NB time_series goodness-of-fit mixing statistics empirical_processes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:8b738190090b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:goodness-of-fit"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:mixing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://projecteuclid.org/euclid.aos/1564797857">
    <title>Tan , Zhang : Doubly penalized estimation in additive regression with high-dimensional data</title>
    <dc:date>2019-08-03T19:29:04+00:00</dc:date>
    <link>https://projecteuclid.org/euclid.aos/1564797857</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Additive regression provides an extension of linear regression by modeling the signal of a response as a sum of functions of covariates of relatively low complexity. We study penalized estimation in high-dimensional nonparametric additive regression where functional semi-norms are used to induce smoothness of component functions and the empirical L2L2 norm is used to induce sparsity. The functional semi-norms can be of Sobolev or bounded variation types and are allowed to be different amongst individual component functions. We establish oracle inequalities for the predictive performance of such methods under three simple technical conditions: a sub-Gaussian condition on the noise, a compatibility condition on the design and the functional classes under consideration and an entropy condition on the functional classes. For random designs, the sample compatibility condition can be replaced by its population version under an additional condition to ensure suitable convergence of empirical norms. In homogeneous settings where the complexities of the component functions are of the same order, our results provide a spectrum of minimax convergence rates, from the so-called slow rate without requiring the compatibility condition to the fast rate under the hard sparsity or certain LqLq sparsity to allow many small components in the true regression function. These results significantly broaden and sharpen existing ones in the literature."]]></description>
<dc:subject>to:NB statistics regression additive_models nonparametrics empirical_processes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:a91a49a02b12/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:regression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:additive_models"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:nonparametrics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1907.09244">
    <title>[1907.09244] Fast rates for empirical risk minimization with cadlag losses with bounded sectional variation norm</title>
    <dc:date>2019-07-24T14:02:23+00:00</dc:date>
    <link>https://arxiv.org/abs/1907.09244</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Empirical risk minimization over sieves of the class  of cadlag functions with bounded variation norm has a long history, starting with Total Variation Denoising (Rudin et al., 1992), and has been considered by several recent articles, in particular Fang et al. (2019) and van der Laan (2015). 
"In this article, we show how a certain representation of cadlag functions with bounded sectional variation, also called Hardy-Krause variation, allows to bound the bracketing entropy of sieves of  and therefore derive fast rates of convergence in nonparametric function estimation. Specifically, for any sequence an that (slowly) diverges to ∞, we show that we can construct an estimator with rate of convergence OP(2d/3n−1/3(logn)d/3a2/3n) over , under some fairly general assumptions. Remarkably, the dimension only affects the rate in n through the logarithmic factor, making this method especially appropriate for high dimensional problems. 
"In particular, we show that in the case of nonparametric regression over sieves of cadlag functions with bounded sectional variation norm, this upper bound on the rate of convergence holds for least-squares estimators, under the random design, sub-exponential errors setting."]]></description>
<dc:subject>to:NB learning_theory method_of_sieves regression empirical_processes statistics van_der_laan.mark</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:50e7cd068af5/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:method_of_sieves"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:regression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:van_der_laan.mark"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://projecteuclid.org/euclid.aos/1558425646">
    <title>Han , Wellner : Convergence rates of least squares regression estimators with heavy-tailed errors</title>
    <dc:date>2019-05-26T22:14:28+00:00</dc:date>
    <link>https://projecteuclid.org/euclid.aos/1558425646</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We study the performance of the least squares estimator (LSE) in a general nonparametric regression model, when the errors are independent of the covariates but may only have a ppth moment (p≥1p≥1). In such a heavy-tailed regression setting, we show that if the model satisfies a standard “entropy condition” with exponent α∈(0,2)α∈(0,2), then the L2L2 loss of the LSE converges at a rate
OP(n−12+α∨n−12+12p).
Such a rate cannot be improved under the entropy condition alone.
"This rate quantifies both some positive and negative aspects of the LSE in a heavy-tailed regression setting. On the positive side, as long as the errors have p≥1+2/αp≥1+2/α moments, the L2L2 loss of the LSE converges at the same rate as if the errors are Gaussian. On the negative side, if p<1+2/αp<1+2/α, there are (many) hard models at any entropy level αα for which the L2L2 loss of the LSE converges at a strictly slower rate than other robust estimators.
"The validity of the above rate relies crucially on the independence of the covariates and the errors. In fact, the L2L2 loss of the LSE can converge arbitrarily slowly when the independence fails.
"The key technical ingredient is a new multiplier inequality that gives sharp bounds for the “multiplier empirical process” associated with the LSE. We further give an application to the sparse linear regression model with heavy-tailed covariates and errors to demonstrate the scope of this new inequality."]]></description>
<dc:subject>to:NB regression empirical_processes statistics heavy_tails nonparametrics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:a745d4cc291e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:regression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:heavy_tails"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:nonparametrics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://projecteuclid.org/euclid.bj/1551862848">
    <title>Lee , Song : Stable limit theorems for empirical processes under conditional neighborhood dependence</title>
    <dc:date>2019-05-25T03:01:30+00:00</dc:date>
    <link>https://projecteuclid.org/euclid.bj/1551862848</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This paper introduces a new concept of stochastic dependence among many random variables which we call conditional neighborhood dependence (CND). Suppose that there are a set of random variables and a set of sigma algebras where both sets are indexed by the same set endowed with a neighborhood system. When the set of random variables satisfies CND, any two non-adjacent sets of random variables are conditionally independent given sigma algebras having indices in one of the two sets’ neighborhood. Random variables with CND include those with conditional dependency graphs and a class of Markov random fields with a global Markov property. The CND property is useful for modeling cross-sectional dependence governed by a complex, large network. This paper provides two main results. The first result is a stable central limit theorem for a sum of random variables with CND. The second result is a Donsker-type result of stable convergence of empirical processes indexed by a class of functions satisfying a certain bracketing entropy condition when the random variables satisfy CND."]]></description>
<dc:subject>to_read empirical_processes random_fields stochastic_processes central_limit_theorem in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:f93b70f7625b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:random_fields"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:central_limit_theorem"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://onlinelibrary.wiley.com/doi/10.1111/jtsa.12256/abstract">
    <title>Block Bootstrap for the Empirical Process of Long-Range Dependent Data - Tewes - 2017 - Journal of Time Series Analysis - Wiley Online Library</title>
    <dc:date>2017-10-12T23:22:20+00:00</dc:date>
    <link>http://onlinelibrary.wiley.com/doi/10.1111/jtsa.12256/abstract</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We consider the bootstrapped empirical process of long-range dependent data. It is shown that this process converges to a semi-degenerate limit, where the random part of this limit is always Gaussian. Thus the bootstrap might fail when the original empirical process accomplishes a noncentral limit theorem. However, even in this case our results can be used to estimate a nuisance parameter that appears in the limit of many nonparametric tests under long memory. Moreover, we develop a new resampling procedure for goodness-of-fit tests and a test for monotonicity of transformations."]]></description>
<dc:subject>stochastic_processes time_series statistics bootstrap empirical_processes long-range_dependence in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:ac0214bb2adb/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:bootstrap"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:long-range_dependence"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://onlinelibrary.wiley.com/doi/10.1111/jtsa.12189/abstract">
    <title>Optimal Rate of Convergence for Empirical Quantiles and Distribution Functions for Time Series - Jirak - 2016 - Journal of Time Series Analysis - Wiley Online Library</title>
    <dc:date>2016-10-18T20:58:30+00:00</dc:date>
    <link>http://onlinelibrary.wiley.com/doi/10.1111/jtsa.12189/abstract</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Given a stationary sequence  , we are interested in the rate of convergence in the central limit theorem of the empirical quantiles and the empirical distribution function. Under a general notion of weak dependence, we show a Berry–Esseen result with optimal rate n−1/2. The setup includes many prominent time series models, such as functions of ARMA or (augmented) GARCH processes. In this context, optimal Berry–Esseen rates for empirical quantiles appear to be novel."]]></description>
<dc:subject>to:NB empirical_processes nonparametrics statistics time_series central_limit_theorem</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:5f76a7a363f4/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:nonparametrics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:central_limit_theorem"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1508.05906">
    <title>[1508.05906] Chaining, Interpolation, and Convexity</title>
    <dc:date>2016-09-07T14:53:37+00:00</dc:date>
    <link>http://arxiv.org/abs/1508.05906</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We show that classical chaining bounds on the suprema of random processes in terms of entropy numbers can be systematically improved when the underlying set is convex: the entropy numbers need not be computed for the entire set, but only for certain "thin" subsets. This phenomenon arises from the observation that real interpolation can be used as a natural chaining mechanism. Unlike the general form of Talagrand's generic chaining method, which is sharp but often difficult to use, the resulting bounds involve only entropy numbers but are nonetheless sharp in many situations in which classical entropy bounds are suboptimal. Such bounds are readily amenable to explicit computations in specific examples, and we discover some old and new geometric principles for the control of chaining functionals as special cases."]]></description>
<dc:subject>empirical_processes learning_theory approximation convexity functional_analysis van_handel.ramon in_NB to_teach:childs_garden_of_statistical_learning_theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:724678402496/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:convexity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:functional_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:van_handel.ramon"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:childs_garden_of_statistical_learning_theory"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1510.04740">
    <title>[1510.04740] Semiparametric theory and empirical processes in causal inference</title>
    <dc:date>2016-09-01T19:20:22+00:00</dc:date>
    <link>http://arxiv.org/abs/1510.04740</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["In this paper we review important aspects of semiparametric theory and empirical processes that arise in causal inference problems. We begin with a brief introduction to the general problem of causal inference, and go on to discuss estimation and inference for causal effects under semiparametric models, which allow parts of the data-generating process to be unrestricted if they are not of particular interest (i.e., nuisance functions). These models are very useful in causal problems because the outcome process is often complex and difficult to model, and there may only be information available about the treatment process (at best). Semiparametric theory gives a framework for benchmarking efficiency and constructing estimators in such settings. In the second part of the paper we discuss empirical process theory, which provides powerful tools for understanding the asymptotic behavior of semiparametric estimators that depend on flexible nonparametric estimators of nuisance functions. These tools are crucial for incorporating machine learning and other modern methods into causal inference analyses. We conclude by examining related extensions and future directions for work in semiparametric causal inference."]]></description>
<dc:subject>to:NB statistics causal_inference nonparametrics empirical_processes to_read kith_and_kin kennedy.edward_h.</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:bb7c068be950/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:causal_inference"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:nonparametrics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kith_and_kin"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kennedy.edward_h."/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1607.06534">
    <title>[1607.06534] The Landscape of Empirical Risk for Non-convex Losses</title>
    <dc:date>2016-07-25T03:15:03+00:00</dc:date>
    <link>http://arxiv.org/abs/1607.06534</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We revisit the problem of learning a noisy linear classifier by minimizing the empirical risk associated to the square loss. While the empirical risk is non-convex, we prove that its structure is remarkably simple. Namely, when the sample size is larger than Cdlogd (with d the dimension, and C a constant) the following happen with high probability: (a) The empirical risk has a unique local minimum (which is also the global minimum); (b) Gradient descent converges exponentially fast to the global minimizer, from any initialization; (c) The global minimizer approaches the true parameter at nearly optimal rate. The core of our argument is to establish a uniform convergence result for the gradients and Hessians of the empirical risk."]]></description>
<dc:subject>learning_theory empirical_processes optimization in_NB high-dimensional_statistics to_teach:childs_garden_of_statistical_learning_theory via:mraginsky</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:bbd8d05b53e2/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:childs_garden_of_statistical_learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:via:mraginsky"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.springer.com/us/book/9783319327730">
    <title>Estimation and Testing Under Sparsity | Sara van de Geer | Springer</title>
    <dc:date>2016-07-06T13:53:56+00:00</dc:date>
    <link>http://www.springer.com/us/book/9783319327730</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Taking the Lasso method as its starting point, this book describes the main ingredients needed to study general loss functions and sparsity-inducing regularizers. It also provides a semi-parametric approach to establishing confidence intervals and tests. Sparsity-inducing methods have proven to be very useful in the analysis of high-dimensional data. Examples include the Lasso and group Lasso methods, and the least squares method with other norm-penalties, such as the nuclear norm. The illustrations provided include generalized linear models, density estimation, matrix completion and sparse principal components. Each chapter ends with a problem section. The book can be used as a textbook for a graduate or PhD course."]]></description>
<dc:subject>to:NB books:noted statistics sparsity high-dimensional_statistics lasso hypothesis_testing confidence_sets van_de_geer.sara to_read empirical_processes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:a21d87b7abc5/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:books:noted"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:sparsity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:lasso"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:hypothesis_testing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:confidence_sets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:van_de_geer.sara"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://link.springer.com/article/10.1007/s11203-015-9120-2?wt_mc=alerts.TOCjournals">
    <title>Blockwise bootstrap of the estimated empirical process based on psi -weakly dependent observations - Springer</title>
    <dc:date>2016-03-14T18:13:05+00:00</dc:date>
    <link>http://link.springer.com/article/10.1007/s11203-015-9120-2?wt_mc=alerts.TOCjournals</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["The distributional convergence of the bootstrapped estimated empirical process is shown and bootstrap consistency in the sup-norm for test statistics based on that process. Bootstrapping the estimated empirical process has up to now been considered by assuming independence of the observations, where we give up this assumption now and allow the observations to be ψ-weakly dependent in the sense of Doukhan and Louhichi (Stoch Proc Appl 84:313–342, 1999). Due to the fact that no model assumptions on the original process are made, a nonparametric blockwise bootstrap procedure is used, which has previously been used in empirical process theory based on mixing observations. Here, we succeeded in proving that assuming l=o(n) and l→∞ as only conditions for the blocklength is sufficient to show convergence of the bootstrap process to the same limit as for the original process under H0, which is the weakest condition that has been imposed in that context up to now."]]></description>
<dc:subject>to:NB empirical_processes bootstrap mixing stochastic_processes statistical_inference_for_stochastic_processes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:5a3cc34601d4/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:bootstrap"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:mixing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistical_inference_for_stochastic_processes"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://projecteuclid.org/euclid.aop/1176989128">
    <title>Arcones , Gine : Limit Theorems for $U$-Processes</title>
    <dc:date>2015-12-09T00:27:53+00:00</dc:date>
    <link>https://projecteuclid.org/euclid.aop/1176989128</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Necessary and sufficient conditions for the law of large numbers and sufficient conditions for the central limit theorem for U-processes are given. These conditions are in terms of random metric entropies. The CLT and LLN for VC subgraph classes of functions as well as for classes satisfying bracketing conditions follow as consequences of the general results. In particular, Liu's simplicial depth process satisfies both the LLN and the CLT. Among the techniques used, randomization, decoupling inequalities, integrability of Gaussian and Rademacher chaos and exponential inequalities for U-statistics should be mentioned."]]></description>
<dc:subject>u-statistics empirical_processes deviation_inequalities vc-dimension central_limit_theorem re:smoothing_adjacency_matrices in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:fbf2713f12b2/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:u-statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:deviation_inequalities"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:vc-dimension"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:central_limit_theorem"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:smoothing_adjacency_matrices"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.springer.com/us/book/9783319218519">
    <title>Measures of Complexity - Festschrift for Alexey Chervonenkis | Vladimir Vovk | Springer</title>
    <dc:date>2015-10-26T03:51:34+00:00</dc:date>
    <link>http://www.springer.com/us/book/9783319218519</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This book brings together historical notes, reviews of research developments, fresh ideas on how to make VC (Vapnik–Chervonenkis) guarantees tighter, and new technical contributions in the areas of machine learning, statistical inference, classification, algorithmic statistics, and pattern recognition."]]></description>
<dc:subject>to:NB learning_theory empirical_processes vc-dimension books:noted to_teach:childs_garden_of_statistical_learning_theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:7a0909ffb330/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:vc-dimension"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:books:noted"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:childs_garden_of_statistical_learning_theory"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.maths.manchester.ac.uk/~goran/lectures.pdf">
    <title>From Uniform Laws of Large Numbers to Uniform Ergodic Theorems</title>
    <dc:date>2015-08-27T00:24:46+00:00</dc:date>
    <link>http://www.maths.manchester.ac.uk/~goran/lectures.pdf</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["The purpose of these lectures is to present three different approaches with their own methods for establishing uniform laws of large numbers and uni- form ergodic theorems for dynamical systems. The presentation follows the principle according to which the i.i.d. case is considered first in great de- tail, and then attempts are made to extend these results to the case of more general dependence structures. The lectures begin (Chapter 1) with a re- view and description of classic laws of large numbers and ergodic theorems, their connection and interplay, and their infinite dimensional extensions to- wards uniform theorems with applications to dynamical systems. The first approach (Chapter 2) is of metric entropy with bracketing which relies upon the Blum-DeHardt law of large numbers and Hoffmann-Jørgensen’s exten- sion of it. The result extends to general dynamical systems using the uniform ergodic lemma (or Kingman’s subadditive ergodic theorem). In this context metric entropy and majorizing measure type conditions are also considered. The second approach (Chapter 3) is of Vapnik and Chervonenkis. It relies upon Rademacher randomization (subgaussian inequality) and Gaussian ran- domization (Sudakov’s minoration) and offers conditions in terms of random entropy numbers. Absolutely regular dynamical systems are shown to sup- port the VC theory using a blocking technique and Eberlein’s lemma. The third approach (Chapter 4) is aimed to cover the wide sense stationary case which is not accessible by the previous two methods. This approach relies upon the spectral representation theorem and offers conditions in terms of the orthogonal stochastic measures which are associated with the underlying dynamical system by means of this theorem. The case of bounded variation is covered, while the case of unbounded variation is left as an open question. The lectures finish with a supplement in which the role of uniform conver- gence of reversed martingales towards consistency of statistical models is explained via the concept of Hardy’s regular convergence."

--- I got a glance at this once a decade ago in a library, and have been looking for a copy for years.

---ETA after reading: [http://bactra.org/weblog/algae-2019-05.html#peskir]]]></description>
<dc:subject>in_NB ergodic_theory vc-dimension learning_theory stochastic_processes empirical_processes have_read books:recommended to_teach:childs_garden_of_statistical_learning_theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:f5f9b778314c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:ergodic_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:vc-dimension"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:books:recommended"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:childs_garden_of_statistical_learning_theory"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/euclid.aos/1407420009">
    <title>Chernozhukov , Chetverikov , Kato : Gaussian approximation of suprema of empirical processes</title>
    <dc:date>2015-02-23T07:23:52+00:00</dc:date>
    <link>http://projecteuclid.org/euclid.aos/1407420009</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This paper develops a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the sup-norm. We prove an abstract approximation theorem applicable to a wide variety of statistical problems, such as construction of uniform confidence bands for functions. Notably, the bound in the main approximation theorem is nonasymptotic and the theorem allows for functions that index the empirical process to be unbounded and have entropy divergent with the sample size. The proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors, the proof of which depends on an effective use of Stein’s method for normal approximation, and some new empirical process techniques. We study applications of this approximation theorem to local and series empirical processes arising in nonparametric estimation via kernel and series methods, where the classes of functions change with the sample size and are non-Donsker. Importantly, our new technique is able to prove the Gaussian approximation for the supremum type statistics under weak regularity conditions, especially concerning the bandwidth and the number of series functions, in those examples."

Ungated: http://arxiv.org/abs/1212.6885]]></description>
<dc:subject>empirical_processes to_read in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:4c682f5d4a08/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/euclid.bj/1402488943">
    <title>Dehling , Durieu , Tusche : Approximating class approach for empirical processes of dependent sequences indexed by functions</title>
    <dc:date>2015-01-24T14:13:31+00:00</dc:date>
    <link>http://projecteuclid.org/euclid.bj/1402488943</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We study weak convergence of empirical processes of dependent data (Xi)i≥0, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class  is different from the class of functions f for which we have good properties of the observables (f(Xi))i≥0. We introduce a new bracketing number to measure the size of the index class  which fits this setting. Our results apply to the empirical process of data (Xi)i≥0 satisfying a multiple mixing condition. This includes dynamical systems and Markov chains, if the Perron–Frobenius operator or the Markov operator has a spectral gap, but also extends beyond this class, for example, to ergodic torus automorphisms."]]></description>
<dc:subject>empirical_processes approximation stochastic_processes markov_models dynamical_systems ergodic_theory mixing in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:b57eb1847383/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:markov_models"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dynamical_systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:ergodic_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:mixing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/euclid.bj/1411134452">
    <title>Lederer , van de Geer : New concentration inequalities for suprema of empirical processes</title>
    <dc:date>2015-01-24T14:12:37+00:00</dc:date>
    <link>http://projecteuclid.org/euclid.bj/1411134452</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["While effective concentration inequalities for suprema of empirical processes exist under boundedness or strict tail assumptions, no comparable results have been available under considerably weaker assumptions. In this paper, we derive concentration inequalities assuming only low moments for an envelope of the empirical process. These concentration inequalities are beneficial even when the envelope is much larger than the single functions under consideration."]]></description>
<dc:subject>empirical_processes concentration_of_measure deviation_inequalities van_de_geer.sara stochastic_processes to_read in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:0877957415cc/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:concentration_of_measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:deviation_inequalities"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:van_de_geer.sara"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/euclid.bj/1411134447">
    <title>Trashorras , Wintenberger : Large deviations for bootstrapped empirical measures</title>
    <dc:date>2015-01-24T14:10:22+00:00</dc:date>
    <link>http://projecteuclid.org/euclid.bj/1411134447</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We investigate the Large Deviations (LD) properties of bootstrapped empirical measures with exchangeable weights. Our main results show in great generality how the resulting rate functions combine the LD properties of both the sample weights and the observations. As an application, we obtain new LD results and discuss both conditional and unconditional LD-efficiency for many classical choices of entries such as Efron’s, leave-p-out, i.i.d. weighted, k-blocks bootstraps, etc."]]></description>
<dc:subject>bootstrap empirical_processes large_deviations stochastic_processes statistics re:almost_none in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:d04a91aae66c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:bootstrap"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:large_deviations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:almost_none"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1409.2090">
    <title>[1409.2090] On the asymptotics of random forests</title>
    <dc:date>2015-01-20T14:07:18+00:00</dc:date>
    <link>http://arxiv.org/abs/1409.2090</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["The last decade has witnessed a growing interest in random forest models which are recognized to exhibit good practical performance, especially in high-dimensional settings. On the theoretical side, however, their predictive power remains largely unexplained, thereby creating a gap between theory and practice. The aim of this paper is twofold. Firstly, we provide theoretical guarantees to link finite forests used in practice (with a finite number M of trees) to their asymptotic counterparts. Using empirical process theory, we prove a uniform central limit theorem for a large class of random forest estimates, which holds in particular for Breiman's original forests. Secondly, we show that infinite forest consistency implies finite forest consistency and thus, we state the consistency of several infinite forests. In particular, we prove that q quantile forests---close in spirit to Breiman's forests but easier to study---are able to combine inconsistent trees to obtain a final consistent prediction, thus highlighting the benefits of random forests compared to single trees."]]></description>
<dc:subject>to:NB decision_trees ensemble_methods empirical_processes statistics random_forests</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:1fb865de58fb/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:decision_trees"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:ensemble_methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:random_forests"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1409.8557">
    <title>[1409.8557] Statistical Theory for High-Dimensional Models</title>
    <dc:date>2015-01-20T02:01:08+00:00</dc:date>
    <link>http://arxiv.org/abs/1409.8557</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["These lecture notes consist of three chapters. In the first chapter we present oracle inequalities for the prediction error of the Lasso and square-root Lasso and briefly describe the scaled Lasso. In the second chapter we establish asymptotic linearity of a de-sparsified Lasso. This implies asymptotic normality under certain conditions and therefore can be used to construct confidence intervals for parameters of interest. A similar line of reasoning can be invoked to derive bounds in sup-norm for the Lasso and asymptotic linearity of de-sparsified estimators of a precision matrix. In the third chapter we consider chaining and the more general generic chaining method developed by Talagrand. This allows one to bound suprema of random processes. Concentration inequalities are refined probability inequalities, mostly again for suprema of random processes. We combine the two. We prove a deviation inequality directly using (generic) chaining."]]></description>
<dc:subject>statistics high-dimensional_statistics concentration_of_measure empirical_processes regression lasso van_de_geer.sara in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:4290f1c88bdc/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:concentration_of_measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:regression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:lasso"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:van_de_geer.sara"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/math/0410072">
    <title>[math/0410072] Higher criticism for detecting sparse heterogeneous mixtures</title>
    <dc:date>2014-10-18T01:04:15+00:00</dc:date>
    <link>http://arxiv.org/abs/math/0410072</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Higher criticism, or second-level significance testing, is a multiple-comparisons concept mentioned in passing by Tukey. It concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis. Tukey suggested comparing the fraction of observed significances at a given \alpha-level to the expected fraction under the joint null. In fact, he suggested standardizing the difference of the two quantities and forming a z-score; the resulting z-score tests the significance of the body of significance tests. We consider a generalization, where we maximize this z-score over a range of significance levels 0<\alpha\leq\alpha_0. 
"We are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem: testing whether n normal means are all zero versus the alternative that a small fraction is nonzero. The subtlety of this ``sparse normal means'' testing problem can be seen from work of Ingster and Jin, who studied such problems in great detail. In their studies, they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p-values either for the bulk of the tests or for the few most highly significant tests. 
"In this range, when the amplitude of nonzero means is calibrated with the fraction of nonzero means, the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses."

--- It makes a lot more sense that the name would come from someone like Tukey.]]></description>
<dc:subject>to:NB multiple_testing hypothesis_testing empirical_processes statistics donoho.david jin.jiashun tukey.john_w. have_read re:network_differences to:blog</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:1877e29a7c8c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:multiple_testing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:hypothesis_testing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:donoho.david"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:jin.jiashun"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:tukey.john_w."/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:network_differences"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:blog"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/euclid.aos/1018031108">
    <title>Koul , Stute : Nonparametric model checks for time series</title>
    <dc:date>2014-08-07T16:09:12+00:00</dc:date>
    <link>http://projecteuclid.org/euclid.aos/1018031108</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This paper studies a class of tests useful for testing the goodness-of-fit of an autoregressive model. These tests are based on a class of empirical processes marked by certain residuals. The paper first gives their large sample behavior under null hypotheses. Then a martingale transformation of the underlying process is given that makes tests based on it asymptotically distribution free. Consistency of these tests is also discussed briefly."]]></description>
<dc:subject>statistics goodness-of-fit time_series empirical_processes in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:603b42bac578/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:goodness-of-fit"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.princeton.edu/~rvan/ORF570.pdf">
    <title>Probability in High Dimension</title>
    <dc:date>2014-07-09T13:26:22+00:00</dc:date>
    <link>https://www.princeton.edu/~rvan/ORF570.pdf</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA[2014 lecture notes for van Handel's class.  Looks great.]]></description>
<dc:subject>concentration_of_measure empirical_processes probability high-dimensional_probability learning_theory vc-dimension van_handel.ramon via:arsyed re:almost_none in_NB to_teach:childs_garden_of_statistical_learning_theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:f6d6eb8bb7ee/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:concentration_of_measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:probability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_probability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:vc-dimension"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:van_handel.ramon"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:via:arsyed"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:almost_none"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:childs_garden_of_statistical_learning_theory"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1401.3034">
    <title>[1401.3034] Inference for Monotone Trends Under Dependence</title>
    <dc:date>2014-03-10T18:02:18+00:00</dc:date>
    <link>http://arxiv.org/abs/1401.3034</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We focus on the problem estimating a monotone trend function under additive and dependent noise. New point-wise confidence interval estimators under both short- and long-range dependent errors are introduced and studied. These intervals are obtained via the method of inversion of certain discrepancy statistics arising in hypothesis testing problems. The advantage of this approach is that it avoids the estimation of nuisance parameters such as the derivative of the unknown function, which existing methods are forced to deal with. While the methodology is motivated by earlier work in the independent context, the dependence of the errors, especially longrange dependence leads to new challenges, such as the study of convex minorants of drifted fractional Brownian motion that may be of independent interest. We also unravel a new family of universal limit distributions (and tabulate selected quantiles) that can henceforth be used for inference in monotone function problems involving dependence."]]></description>
<dc:subject>to:NB nonparametrics regression statistics empirical_processes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:356ff9f12790/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:nonparametrics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:regression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://link.springer.com/article/10.1007/s10959-012-0450-3?wt_mc=alerts.TOCjournals">
    <title>An Empirical Process Central Limit Theorem for Multidimensional Dependent Data - Springer</title>
    <dc:date>2014-03-10T15:11:23+00:00</dc:date>
    <link>http://link.springer.com/article/10.1007/s10959-012-0450-3?wt_mc=alerts.TOCjournals</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Let (Un(t))t∈ℝd be the empirical process associated to an ℝ d -valued stationary process (X i ) i≥0. In the present paper, we introduce very general conditions for weak convergence of (Un(t))t∈ℝd , which only involve properties of processes (f(X i )) i≥0 for a restricted class of functions f∈ . Our results significantly improve those of Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011) and provide new applications.
"The central interest in our approach is that it does not need the indicator functions which define the empirical process (Un(t))t∈ℝd to belong to the class   . This is particularly useful when dealing with data arising from dynamical systems or functionals of Markov chains. In the proofs we make use of a new application of a chaining argument and generalize ideas first introduced in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011).
"Finally we will show how our general conditions apply in the case of multiple mixing processes of polynomial decrease and causal functions of independent and identically distributed processes, which could not be treated by the preceding results in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011)."]]></description>
<dc:subject>empirical_processes stochastic_processes mixing re:almost_none in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:143a1f4e33e0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:mixing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:almost_none"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1402.2718">
    <title>[1402.2718] Concentration of random polytopes around the expected convex hull</title>
    <dc:date>2014-02-18T00:38:09+00:00</dc:date>
    <link>http://arxiv.org/abs/1402.2718</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We provide a streamlined proof and improved estimates for the weak multivariate Gnedenko law of large numbers on concentration of random polytopes within the space of convex bodies (in a fixed or a high dimensional setting), as well as a corresponding strong law of large numbers."]]></description>
<dc:subject>empirical_processes probability high-dimensional_probability in_NB geometry</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:c473014dfaa8/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:probability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_probability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:geometry"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1402.2918">
    <title>[1402.2918] Confidence Bands for Distribution Functions: A New Look at the Law of the Iterated Logarithm</title>
    <dc:date>2014-02-13T18:22:03+00:00</dc:date>
    <link>http://arxiv.org/abs/1402.2918</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We present a general law of the iterated logarithm for stochastic processes on the open unit interval having subexponential tails in a locally uniform fashion. It applies to standard Brownian bridge but also to suitably standardized empirical distribution functions. This leads to new goodness-of-fit tests and confidence bands which refine the procedures of Berk and Jones (1979) and Owen (1995). Roughly speaking, the high power and accuracy of the latter procedures in the tail regions of distributions are esentially preserved while gaining considerably in the central region."

- Relevant for updating Clauset's procedure for picking xmin in a power law?]]></description>
<dc:subject>empirical_processes statistics confidence_sets re:pli-R in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:bdea1795ab46/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:confidence_sets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:pli-R"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1312.1005">
    <title>[1312.1005] Squared-Norm Empirical Process in Banach Space</title>
    <dc:date>2014-01-02T17:54:55+00:00</dc:date>
    <link>http://arxiv.org/abs/1312.1005</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This note extends a recent result of Mendelson on the supremum of a quadratic process to squared norms of functions taking values in a Banach space. Our method of proof is a reduction by a symmetrization argument and observation about the subadditivity of the generic chaining functional. We provide an application to the supremum of a linear process in the sample covariance matrix indexed by finite rank, positive definite matrices."]]></description>
<dc:subject>empirical_processes kith_and_kin vu.vince lei.jing in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:75dfd1030d20/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kith_and_kin"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:vu.vince"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:lei.jing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1312.5894">
    <title>[1312.5894] Weak Convergence of the Sequential Empirical Process of some Long-Range Dependent Sequences with Respect to a Weighted Norm</title>
    <dc:date>2013-12-26T17:16:17+00:00</dc:date>
    <link>http://arxiv.org/abs/1312.5894</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Let (Xk)k≥1 be a Gaussian long-range dependent process with EX1=0, EX21=1 and covariance function r(k)=k−DL(k). For any measurable function G let (Yk)k≥1=(G(Xk))k≥1. We study the asymptotic behaviour of the associated sequential empirical process (RN(x,t)) with respect to a weighted norm ∥⋅∥w. We show that, after an appropriate normalization, (RN(x,t)) converges weakly in the space of c\`adl\`ag functions with finite weighted norm to a Hermite process."]]></description>
<dc:subject>empirical_processes stochastic_processes long-range_dependence in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:caac9d6b56f0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:long-range_dependence"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.bj/1386078618">
    <title>Preuß , Vetter , Dette : A test for stationarity based on empirical processes</title>
    <dc:date>2013-12-03T15:55:03+00:00</dc:date>
    <link>http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.bj/1386078618</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["In this paper we investigate the problem of testing the assumption of stationarity in locally stationary processes. The test is based on an estimate of a Kolmogorov–Smirnov type distance between the true time varying spectral density and its best approximation through a stationary spectral density. Convergence of a time varying empirical spectral process indexed by a class of certain functions is proved, and furthermore the consistency of a bootstrap procedure is shown which is used to approximate the limiting distribution of the test statistic. Compared to other methods proposed in the literature for the problem of testing for stationarity the new approach has at least two advantages: On one hand, the test can detect local alternatives converging to the null hypothesis at any rate gT→0 such that gTT1/2→∞, where T denotes the sample size. On the other hand, the estimator is based on only one regularization parameter while most alternative procedures require two. Finite sample properties of the method are investigated by means of a simulation study, and a comparison with several other tests is provided which have been proposed in the literature."

Ungated: http://arxiv.org/abs/1312.5448]]></description>
<dc:subject>to:NB empirical_processes stochastic_processes statistical_inference_for_stochastic_processes hypothesis_testing statistics non-stationarity</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:2d8ed594c1dd/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistical_inference_for_stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:hypothesis_testing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:non-stationarity"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1310.5796">
    <title>[1310.5796] Relative Deviation Learning Bounds and Generalization with Unbounded Loss Functions</title>
    <dc:date>2013-10-23T14:15:38+00:00</dc:date>
    <link>http://arxiv.org/abs/1310.5796</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We present an extensive analysis of relative deviation bounds, including detailed proofs of two-sided inequalities and their implications. We also give detailed proofs of two-sided generalization bounds that hold in the general case of unbounded loss functions, under the assumption that a moment of the loss is bounded. These bounds are useful in the analysis of importance weighting and other learning tasks such as unbounded regression."]]></description>
<dc:subject>learning_theory vc-dimension empirical_processes have_read in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:4e3b59d92bd3/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:vc-dimension"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1310.5523">
    <title>[1310.5523] On the uniform convergence of empirical norms and inner products, with application to causal inference</title>
    <dc:date>2013-10-23T14:11:54+00:00</dc:date>
    <link>http://arxiv.org/abs/1310.5523</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Uniform convergence of empirical norms - empirical measures of squared functions - is a topic which has received considerable attention in the literature on empirical processes. The results are relevant as empirical norms occur due to symmetrization. They also play a prominent role in statistical applications. The contraction inequality has been a main tool but recently other approaches have shown to lead to better results in important cases. We present an overview including the linear (anisotropic) case, and give new results for inner products of functions. Our main application will be the estimation of the parental structure in a directed acyclic graph. As intermediate result we establish convergence of the least squares estimator when the model is wrong."]]></description>
<dc:subject>concentration_of_measure empirical_processes causal_inference statistics nonparametrics van_de_geer.sara to_read entableted in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:6c426a1eb91c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:concentration_of_measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:causal_inference"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:nonparametrics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:van_de_geer.sara"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:entableted"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://link.springer.com/article/10.1007/s00440-012-0455-y">
    <title>The Bernstein–Orlicz norm and deviation inequalities - Springer</title>
    <dc:date>2013-09-19T15:29:01+00:00</dc:date>
    <link>http://link.springer.com/article/10.1007/s00440-012-0455-y</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein–Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining and generic chaining. The supremum of the empirical process is then studied as a special case. We show that chaining along a tree can be done using entropy with bracketing. Finally, we establish a deviation inequality for the empirical process for the unbounded case."

--- Ungated: http://arxiv.org/abs/1111.2450]]></description>
<dc:subject>empirical_processes deviation_inequalities van_de_geer.sara in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:de384e3fe4bc/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:deviation_inequalities"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:van_de_geer.sara"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://onlinelibrary.wiley.com/doi/10.1111/sjos.12030/abstract">
    <title>Testing for a Change of the Innovation Distribution in Nonparametric Autoregression: The Sequential Empirical Process Approach - Selk - 2013 - Scandinavian Journal of Statistics - Wiley Online Library</title>
    <dc:date>2013-09-17T22:28:10+00:00</dc:date>
    <link>http://onlinelibrary.wiley.com/doi/10.1111/sjos.12030/abstract</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We consider a nonparametric autoregression model under conditional heteroscedasticity with the aim to test whether the innovation distribution changes in time. To this end, we develop an asymptotic expansion for the sequential empirical process of nonparametrically estimated innovations (residuals). We suggest a Kolmogorov–Smirnov statistic based on the difference of the estimated innovation distributions built from the first ⌊ns⌋and the last n − ⌊ns⌋ residuals, respectively (0 ≤ s ≤ 1). Weak convergence of the underlying stochastic process to a Gaussian process is proved under the null hypothesis of no change point. The result implies that the test is asymptotically distribution-free. Consistency against fixed alternatives is shown. The small sample performance of the proposed test is investigated in a simulation study and the test is applied to a data example."]]></description>
<dc:subject>to:NB time_series change-point_problem nonparametrics regression statistics empirical_processes hypothesis_testing</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:72096b56b6e7/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:change-point_problem"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:nonparametrics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:regression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:hypothesis_testing"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aop/1378991852">
    <title>Dedecker , Merlevède , Rio : Strong approximation results for the empirical process of stationary sequences</title>
    <dc:date>2013-09-12T20:03:08+00:00</dc:date>
    <link>http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aop/1378991852</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We prove a strong approximation result for the empirical process associated to a stationary sequence of real-valued random variables, under dependence conditions involving only indicators of half lines. This strong approximation result also holds for the empirical process associated to iterates of expanding maps with a neutral fixed point at zero, as soon as the correlations decrease more rapidly than n−1−δ for some positive δ. This shows that our conditions are in some sense optimal."

Ungated: http://arxiv.org/abs/1310.5451]]></description>
<dc:subject>to:NB empirical_processes stochastic_processes dynamical_systems</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:420cc1ef8399/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dynamical_systems"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1212.6906">
    <title>[1212.6906] Central Limit Theorems and Multiplier Bootstrap when p is much larger than n</title>
    <dc:date>2013-09-09T03:36:11+00:00</dc:date>
    <link>http://arxiv.org/abs/1212.6906</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We derive a central limit theorem for the maximum of a sum of high dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. The key innovation of this result is that it applies even when the dimension of random vectors (p) is large compared to the sample size (n); in fact, p can be much larger than n. We also show that the distribution of the maximum of a sum of the random vectors with unknown covariance matrices can be consistently estimated by the distribution of the maximum of a sum of the conditional Gaussian random vectors obtained by multiplying the original vectors with i.i.d. Gaussian multipliers. This is the multiplier bootstrap procedure. Here too, p can be large or even much larger than n. These distributional approximations, either Gaussian or conditional Gaussian, yield a high-quality approximation to the distribution of the original maximum, often with approximation error decreasing polynomially in the sample size, and hence are of interest in many applications. We demonstrate how our central limit theorem and the multiplier bootstrap can be used for high dimensional estimation, multiple hypothesis testing, and adaptive specification testing. All these results contain non-asymptotic bounds on approximation errors. "

- For the reading group this week.]]></description>
<dc:subject>to:NB empirical_processes stochastic_processes approximation central_limit_theorem bootstrap statistics probability</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:038f7e73d1e5/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:central_limit_theorem"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:bootstrap"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:probability"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://link.springer.com/article/10.1007/s10959-011-0376-1">
    <title>Uniform-in-Bandwidth Functional Limit Laws - Springer</title>
    <dc:date>2013-08-17T19:00:06+00:00</dc:date>
    <link>http://link.springer.com/article/10.1007/s10959-011-0376-1</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We provide uniform-in-bandwidth functional limit laws for the increments of the empirical and quantile processes. Our theorems, established in the framework of convergence in probability, imply new sharp uniform-in-bandwidth limit laws for functional estimators. In particular, they yield the explicit value of the asymptotic limiting constant for the uniform-in-bandwidth sup-norm of the random error of kernel density estimators. We allow the bandwidth to vary within the complete range for which the estimators are consistent."]]></description>
<dc:subject>empirical_processes density_estimation statistics convergence_of_stochastic_processes in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:95b3eab71ec7/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:density_estimation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:convergence_of_stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1307.1565">
    <title>[1307.1565] Concentration inequalities for smooth random fields</title>
    <dc:date>2013-07-08T16:32:27+00:00</dc:date>
    <link>http://arxiv.org/abs/1307.1565</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["In this note we derive a sharp concentration inequality for the supremum of a smooth random field over a finite dimensional set. It is shown that this supremum can be bounded with high probability by the value of the field at some deterministic point plus an intrinsic dimension of the optimisation problem. As an application we prove the exponential inequality for a function of the maximal eigenvalue of a random matrix is proved."]]></description>
<dc:subject>random_fields empirical_processes concentration_of_measure stochastic_processes re:almost_none in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:145934415469/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:random_fields"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:concentration_of_measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:almost_none"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1305.6408">
    <title>[1305.6408] When uniform weak convergence fails: empirical processes for dependence functions via epi- and hypographs</title>
    <dc:date>2013-05-29T13:44:24+00:00</dc:date>
    <link>http://arxiv.org/abs/1305.6408</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["For copulas whose partial derivatives are not continuous everywhere on the interior of the unit cube, the empirical copula process does not converge weakly with respect to the supremum distance. This makes it hard to verify asymptotic properties of inference procedures for such copulas. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epi- and hypoconvergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies $L^p$ convergence. Even in cases where uniform convergence fails, weak convergence with respect to the new metric is established for empirical copula and tail dependence processes. No additional assumptions are needed for tail dependence functions, and for copulas, the assumptions reduce to existence and continuity of the partial derivatives almost everywhere on the unit cube. The results are applied to obtain asymptotic properties of minimum distance estimators, goodness-of-fit tests and resampling procedures."]]></description>
<dc:subject>convergence_of_stochastic_processes empirical_processes probability statistics copulas in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:d90fd135f357/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:convergence_of_stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:probability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:copulas"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1305.5618">
    <title>[1305.5618] A general approach to the joint asymptotic analysis of statistics from sub-samples</title>
    <dc:date>2013-05-27T12:15:21+00:00</dc:date>
    <link>http://arxiv.org/abs/1305.5618</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["In time series analysis, statistics based on collections of estimators computed from sub-samples play a crucial role in an increasing variety of important applications. Proving results about the joint asymptotic distribution of such statistics is challenging since it typically involves a nontrivial verification of technical conditions and tedious case-by-case asymptotic analysis. In this paper, we provide a novel technique that allows to circumvent those problems in a general setting. Our approach consists of two major steps: a probabilistic part which is mainly concerned with weak convergence of sequential empirical processes, and an analytic part providing general ways to extend this weak convergence to functionals of the sequential empirical process. Our theory provides a unified treatment of asymptotic distributions for a large class of statistics, including recently proposed self-normalized statistics and sub-sampling based p-values. In addition, we comment on the consistency of bootstrap procedures and obtain general results on compact differentiability of certain mappings that seem to be of independent interest."]]></description>
<dc:subject>to_read re:XV_for_mixing empirical_processes bootstrap time_series convergence_of_stochastic_processes statistical_inference_for_stochastic_processes statistics in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:dac13484b7b7/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:XV_for_mixing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:bootstrap"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:convergence_of_stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistical_inference_for_stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1304.5113">
    <title>[1304.5113] A note on weak convergence of the sequential multivariate empirical process under strong mixing</title>
    <dc:date>2013-04-22T17:23:25+00:00</dc:date>
    <link>http://arxiv.org/abs/1304.5113</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This article investigates weak convergence of the sequential $d$-dimensional empirical process under strong mixing. Weak convergence is established for mixing rates $\alpha_n = O(n^{-a})$, where $a>1$, which slightly improves upon existing results in the literature that are based on mixing rates depending on the dimension $d$."]]></description>
<dc:subject>to:NB mixing ergodic_theory empirical_processes stochastic_processes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:c7b219da085f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:mixing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:ergodic_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/math/0512554">
    <title>[math/0512554] On weakly bounded empirical processes</title>
    <dc:date>2013-04-13T22:13:37+00:00</dc:date>
    <link>http://arxiv.org/abs/math/0512554</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Let $F$ be a class of functions on a probability space $(\Omega,\mu)$ and let $X_1,...,X_k$ be independent random variables distributed according to $\mu$. We establish high probability tail estimates of the form $\sup_{f \in F} |\{i : |f(X_i)| \geq t \}$ using a natural parameter associated with $F$. We use this result to analyze weakly bounded empirical processes indexed by $F$ and processes of the form $Z_f=|k^{-1}\sum_{i=1}^k |f|^p(X_i)-\E|f|^p|$ for $p>1$. We also present some geometric applications of this approach, based on properties of the random operator $\Gamma=k^{-1/2}\sum_{i=1}^k \inr{X_i,\cdot}e_i$, where the $(X_i)_{i=1}^k$ are sampled according to an isotropic, log-concave measure on $\R^n$."]]></description>
<dc:subject>empirical_processes in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:64eeea38042e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1303.4537">
    <title>[1303.4537] A Sequential Empirical Central Limit Theorem for Multiple Mixing Processes with Application to B-Geometrically Ergodic Markov Chains</title>
    <dc:date>2013-03-20T01:56:30+00:00</dc:date>
    <link>http://arxiv.org/abs/1303.4537</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We investigate the convergence in distribution of sequential empirical processes of dependent data indexed by a class of functions F. Our technique is suitable for processes that satisfy a multiple mixing condition on a space of functions which differs from the class F. This situation occurs in the case of data arising from dynamical systems or Markov chains, for which the Perron--Frobenius or Markov operator, respectively, has a spectral gap on a restricted space. We provide applications to iterative Lipschitz models that contract on average."]]></description>
<dc:subject>to:NB stochastic_processes empirical_processes mixing ergodic_theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:6d870f677691/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:mixing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:ergodic_theory"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aop/1362750942">
    <title>Kuelbs , Kurtz , Zinn : A CLT for empirical processes involving time-dependent data</title>
    <dc:date>2013-03-10T01:14:54+00:00</dc:date>
    <link>http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aop/1362750942</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["For stochastic processes {Xt : t∈E}, we establish sufficient conditions for the empirical process based on {IXt≤y−Pr(Xt≤y) : t∈E,y∈ℝ} to satisfy the CLT uniformly in t∈E, y∈ℝ. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and E=[0,1]."]]></description>
<dc:subject>to:NB empirical_processes stochastic_processes central_limit_theorem kurtz.thomas_g.</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:9da8e56eddc3/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:central_limit_theorem"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kurtz.thomas_g."/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1212.6885">
    <title>[1212.6885] Gaussian approximation of suprema of empirical processes</title>
    <dc:date>2013-01-07T23:08:30+00:00</dc:date>
    <link>http://arxiv.org/abs/1212.6885</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We develop a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating empirical processes themselves in the sup-norm. We prove an abstract approximation theorem that is applicable to a wide variety of problems, primarily in statistics. Especially, the bound in the main approximation theorem is non-asymptotic and the theorem does not require uniform boundedness of the class of functions. The proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors, the proof of which depends on an effective use of Stein's method for normal approximation, and some new empirical processes techniques. We study applications of this approximation theorem to local empirical processes and series estimation in nonparametric regression where the classes of functions change with the sample size and are not Donsker-type. Importantly, our new technique is able to prove the Gaussian approximation for the supremum type statistics under considerably weak regularity conditions, especially concerning the bandwidth and the number of series functions, in those examples."]]></description>
<dc:subject>stochastic_processes empirical_processes learning_theory re:almost_none in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:2efec7591cb9/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:almost_none"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9469.2012.00820.x/abstract">
    <title>Testing Monotonicity of Regression Functions – An Empirical Process Approach - BIRKE - 2012 - Scandinavian Journal of Statistics - Wiley Online Library</title>
    <dc:date>2012-12-18T14:02:04+00:00</dc:date>
    <link>http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9469.2012.00820.x/abstract</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We propose several new tests for monotonicity of regression functions based on different empirical processes of residuals and pseudo-residuals. The residuals are obtained from an unconstrained kernel regression estimator whereas the pseudo-residuals are obtained from an increasing regression estimator. Here, in particular, we consider a recently developed simple kernel-based estimator for increasing regression functions based on increasing rearrangements of unconstrained non-parametric estimators. The test statistics are estimated distance measures between the regression function and its increasing rearrangement. We discuss the asymptotic distributions, consistency and small sample performances of the tests."]]></description>
<dc:subject>to:NB to_read regression statistics model_checking to_teach:undergrad-ADA empirical_processes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:b28a508a7443/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:regression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:model_checking"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:undergrad-ADA"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.bj/1352727815">
    <title>Bonnéry , Breidt , Coquet : Uniform convergence of the empirical cumulative distribution function under informative selection from a finite population</title>
    <dc:date>2012-11-13T21:56:57+00:00</dc:date>
    <link>http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.bj/1352727815</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Consider informative selection of a sample from a finite population. Responses are realized as independent and identically distributed (i.i.d.) random variables with a probability density function (p.d.f.) f, referred to as the superpopulation model. The selection is informative in the sense that the sample responses, given that they were selected, are not i.i.d. f. In general, the informative selection mechanism may induce dependence among the selected observations. The impact of such dependence on the empirical cumulative distribution function (c.d.f.) is studied. An asymptotic framework and weak conditions on the informative selection mechanism are developed under which the (unweighted) empirical c.d.f. converges uniformly, in L2 and almost surely, to a weighted version of the superpopulation c.d.f. This yields an analogue of the Glivenko–Cantelli theorem. A series of examples, motivated by real problems in surveys and other observational studies, shows that the conditions are verifiable for specified designs."]]></description>
<dc:subject>to:NB learning_theory statistics empirical_processes survey_sampling</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:35e97456a76a/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:empirical_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:survey_sampling"/>
</rdf:Bag></taxo:topics>
</item>
</rdf:RDF>