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Persistence of plug-in rule in classification of high dimensional multivariate binary data

JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2007年第11期137卷 3687-3705页

作者： Park, Junyong Ghosh, Jayanta K. Univ Maryland Dept Math & Stat Baltimore Country MD 21250 USA Indian Stat Inst Div Theoret Stat & Math Bangalore Karnataka India

In this paper. we consider the classification problem when the predictors are multivariate binary random variables. Variables are modeled as independent. but not necessarily identical, Bernoulli. A triangular array for parameters. (p(11)((n)) ..., p(1d)((n)), p(21)((n)) ..., p(2d)((n))), is assumed to allow parameters to change and the number of the variables, d, to increase for adopting more flexible models as the sample size. it. increases. Our results are obtained under moderate assumptions on the triangular array of the probability vectors. We use maximum likelihood estimators for the parameters and plug them into the Bayes classifier. This is a plug-in classifier, a sort of objective Bayes rule. It is shown in Wilbur et a]. [2002. Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial DNA fingerprints. Biometrics 58, 378-386] via simulations that the plug-in rule classifies quite well even when the assumption of independence is violated. The main interest in this paper is in the complex case of d/n(v) -> c for some v > 0 and c > 0 tor which very little is known. Using linearity of the plug-in rule, we show its persistence, a generalization of the notion of consistency, when the variance of the plug-in rule or a quantity measuring signal to noise ratio is divergent otherwise we show there exists an in example of non-persistence of the plug-in rule. In case of non-persistence, we introduce the notion of sparsity and overcome non-persistence by selecting a subset of the variables. This shows why a variable selection procedure may he effective especially for contemporary practical problems with high dimensional data [Wilbur et al, 2002. Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial DNA *** 58, 378-386]. (C) 2007 Elsevier B.V. All rights reserved.

关键词： persistence triangular arrays high dimensional multivariate binary data plug-in ruled sparsity

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Consistency of adaptive importance sampling and recycling schemes

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BERNOULLI 2019年第3期25卷 1977-1998页

作者： Marin, Jean-Michel Pudlo, Pierre Sedki, Mohammed Univ Montpellier IMAG CNRS Montpellier France Aix Marseille Univ CNRS UMR I2M 7373 Cent Marseille F-13453 Marseille 13 France Univ Paris Sud U1181 Hop Paul Brousse Bat Inserm 15-1616 Ave Paul Vaillant Couturier F-94807 Villejuif France

Among Monte Carlo techniques, the importance sampling requires fine tuning of a proposal distribution, which is now fluently resolved through iterative schemes. Sequential adaptive algorithms have been proposed to calibrate the sampling distribution. Cornuet et al. [Scand. J. Stat. 39 (2012) 798-812] provides a significant improvement in stability and effective sample size by the introduction of a recycling procedure. However, the consistency of such algorithms have been rarely tackled because of their complexity. Moreover, the recycling strategy of the AMIS estimator adds another difficulty and its consistency remains largely open. In this work, we prove the convergence of sequential adaptive sampling, with finite Monte Carlo sample size at each iteration, and consistency of recycling procedures. Contrary to Douc et al. [Ann. Statist. 35 (2007) 420-448], results are obtained here in the asymptotic regime where the number of iterations is going to infinity while the number of drawings per iteration is a fixed, but growing sequence of integers. Hence, some of the results shed new light on adaptive population Monte Carlo algorithms in that last regime and give advices on how the sample sizes should be fixed.

关键词： adaptive algorithms importance sampling Monte Carlo methods population Monte Carlo sequential Monte Carlo triangular arrays

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Maxima of independent, non-identically distributed Gaussian vectors

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BERNOULLI 2015年第1期21卷 38-61页

作者： Engelke, Sebastian Kabluchko, Zakhar Schlather, Martin Univ Gottingen Inst Math Stochast D-37077 Gottingen Germany Univ Lausanne Fac Hautes Etudes Commerciales Extranef UNIL Dorigny CH-1015 Lausanne Switzerland Univ Ulm Inst Stochast D-89069 Ulm Germany Univ Mannheim Inst Math D-68131 Mannheim Germany

Let X-i,X-n, n is an element of N,1 <= i <= n, be a triangular array of independent R-d-valued Gaussian random vectors with correlation matrices Sigma(i,n). We give necessary conditions under which the row-wise maxima converge to some max-stable distribution which generalizes the class of Husler-Reiss distributions. In the bivariate case, the conditions will also be sufficient. Using these results, new models for bivariate extremes are derived explicitly. Moreover, we define a new class of stationary, max-stable processes as max-mixtures of Brown-Resnick processes. As an application, we show that these processes realize a large set of extremal correlation functions, a natural dependence measure for max-stable processes. This set includes all functions psi (root gamma(h)), h is an element of R-d, where psi is a completely monotone function and gamma is an arbitrary variogram.

关键词： extremal correlation function Gaussian random vectors Husler-Reiss distributions max-limit theorems max-stable distributions triangular arrays

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Necessary and sufficient conditions for the conditional central limit theorem

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ANNALS OF PROBABILITY 2002年第3期30卷 1044-1081页

作者： Dedecker, J Merlevède, F Univ Paris 06 LSTA F-75013 Paris 05 France

Following Lindeberg's approach, we obtain a new condition for a stationary sequence of square-integrable and real-valued random variables to satisfy the central limit theorem. In the adapted case, this condition is weaker than any projective criterion derived from Gordin's theorem [Dokl. Akad. Nauk SSSR 188 (1969) 739-741] about approximating martingales. Moreover, our criterion is equivalent to the conditional central limit theorem, which implies stable convergence (in the sense of Renyi) to a mixture of normal distributions. We also establish functional and triangular versions of this theorem. From these general results, we derive sufficient conditions which are easier to verify and may be compared to other results in the literature. To be complete, we present an application to kernel density estimators for some classes of discrete time processes.

关键词： central limit theorem stable convergence invariance principles stationary processes triangular arrays

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A law of the single logarithm for weighted sums of arrays applied to bootstrap model selection in regression

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STATISTICS & PROBABILITY LETTERS 2012年第5期82卷 965-971页

作者： de Micheaux, Pierre Lafaye Leger, Christian Univ Montreal Dept Math & Stat Montreal PQ H3C 3J7 Canada

We generalize a law of the single logarithm obtained by Qi (1994) and Li et al. (1995) to the case of weighted sums of triangular arrays of random variables. We apply this result to bootstrapping the all-subsets model selection problem in regression, where we show that the popular Bayesian Information Criterion of Schwarz (1978) is no longer asymptotically consistent. (c) 2012 Elsevier B.V. All rights reserved.

关键词： BIC Linear regression Variable selection Rowwise independent triangular arrays

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On wide-(s) sequences and their applications to certain classes of operators

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PACIFIC JOURNAL OF MATHEMATICS 1999年第2期189卷 311-338页

作者： Rosenthal, H Univ Texas Austin TX 78712 USA

A basic sequence in a Banach space is called wide-(s) if it is bounded and dominates the summing basis. (Wide-(s) sequences were originally introduced by I. Singer, who termed them P*-sequences.) These sequences and their quantified versions, termed lambda-wide-(s) sequences, are used to characterize various classes of operators between Banach spaces, such as the weakly compact, Tauberian, and super-Tauberian operators, as well as a new intermediate class introduced here, the strongly Tauberian operators. This is a nonlocalizable class which nevertheless forms an open semigroup and is closed under natural operations such as taking double adjoints. It is proved for example that an operator is non-weakly compact iff for every epsilon > 0, it maps some (1 + epsilon)-wide-(s)-sequence to a wide-(s) sequence. This yields the quantitative triangular arrays result characterizing reflexivity, due to R.C. James. It is shown that an operator is non-Tauberian (resp. non-strongly Tauberian) iff for every epsilon > 0, it maps some (1 + epsilon)-wide-(s) sequence into a norm-convergent sequence (resp. a sequence whose image has diameter less than epsilon). This is applied to obtain a direct "finite" characterization of super-Tauberian operators, as well as the following characterization, which strengthens a recent result of M. Gonzalez and A. Martinez-Abejon: An operator is non-super-Tauberian iff there are for every epsilon > 0, finite (1 + epsilon)-wide-(s) sequences of arbitrary length whose images have norm at most epsilon.

关键词： .weakly compact exivity basic sequence triangular arrays classes of operators WIDE SEQUENCES non-strongly Tauberian

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Extremal limit laws for a class of bivariate Poisson vectors

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STATISTICS & PROBABILITY LETTERS 2001年第4期54卷 373-379页

作者： Coles, S Pauli, F Univ Bristol Sch Math Dept Math Bristol BS8 1TW Avon England Univ Padua Dept Stat Sci Padua Italy

It is well known that conventional extreme value limit laws break down for the Poisson distribution: no normalization can be found to avoid degeneracy of the limit law of sample maxima. Anderson et al. (Ann. Appl. Probab. 7 (1997) 953) tackled this problem with a triangular array argument, letting both the sample size and Poisson mean grow at appropriate rates. This leads to a Gumbel limit law for sample maxima. In applications, this means that it may be appropriate to model extremes of Poisson processes using standard extreme value models and techniques. This paper extends the limit results to a class of bivariate Poisson distributions. Suitably normalized, and with a degree of dependence that is also permitted to grow at a suitable rate, we find that the limit distribution corresponds to the class of bivariate extreme value models that would have arisen, had the population been bivariate normal, cf. Husler and Reiss (Statist. Probab. Lett. 7 (1989) 283). This adds weight to the argument that, for practical applications involving Poisson variables, even in the presence of dependence, standard extreme value models can be applied, despite the degeneracy that arises by applying the usual asymptotic argument, (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： bivariate Poisson extreme value distributions maxima triangular arrays

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Necessary conditions for the bootstrap of the mean of a triangular array

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ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES 1999年第3期35卷 371-386页

作者： del Barrio, E Matrán, C Cuesta-Albertos, JA Univ Valladolid Dept Estadist & Invest Operativa Valladolid Spain Univ Cantabria Dept Matemat Estadist & Computac Santander Spain

Although necessary conditions for the bootstrap of the mean to work have already been given, only the case of an i.i.d. sequence has been exhaustively considered. We study such necessary conditions for (not necessarily infinitesimal) triangular arrays showing that the existence of a limit law in probability leads to infinitesimality and to the Central Limit Theorem to hold for a rescaled subarray. Our setup is based on a triangular array of row-wise independent identically distributed random variables and any resampling size. While our results are similar to those obtained in Arcones and Gine [Ann. Inst. Henri Poincare 25(1989) 457-481] for an i.i.d. sequence, our proof is based on symmetrizations and the consideration of U-statistics and allows a unified treatment without moment assumptions. (C) Elsevier, Paris.

关键词： necessary condition bootstrap mean arbitrary resampling sizes triangular arrays U-statistics Central Limit Theorem

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Limit theorems for empirical R,nyi entropy and divergence with applications to molecular diversity analysis

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TEST 2016年第4期25卷 654-673页

作者： Pietrzak, Maciej Rempala, Grzegorz A. Seweryn, Michal Wesolowski, Jacek Ohio State Univ Coll Publ Hlth Div Biostat Columbus OH 43210 USA Ohio State Univ Math Biosci Inst Columbus OH 43210 USA Univ Lodz Dept Math & Comp Sci Lodz Poland Warsaw Univ Sci & Technol Wydzial Matemat & Nauk Informacyjnych Warsaw Poland

Quantitative methods for studying biodiversity have been traditionally rooted in the classical theory of finite frequency tables analysis. However, with the help of modern experimental tools, like high-throughput sequencing, we now begin to unlock the outstanding diversity of genomic data in plants and animals reflective of the long evolutionary history of our planet. This molecular data often defies the classical frequency/contingency tables assumptions and seems to require sparse tables with very large number of categories and highly unbalanced cell counts, e.g., following heavy-tailed distributions (for instance, power laws). Motivated by the molecular diversity studies, we propose here a frequency-based framework for biodiversity analysis in the asymptotic regime where the number of categories grows with sample size (an infinite contingency table). Our approach is rooted in information theory and based on the Gaussian limit results for the effective number of species (the Hill numbers) and the empirical Renyi entropy and divergence. We argue that when applied to molecular biodiversity analysis, our methods can properly account for the complicated data frequency patterns on one hand and the practical sample size limitations on the other. We illustrate this principle with two specific RNA sequencing examples: a comparative study of T-cell receptor populations and a validation of some preselected molecular hepatocellular carcinoma (HCC) markers.

关键词： Hill number Central limit theorem Next-generation sequencing triangular arrays T-cell receptors

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PRODUCT FORMULAS FOR VOLUMES OF FLOW POLYTOPES

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2015年第3期143卷 937-954页

作者： Meszaros, Karola Cornell Univ Dept Math Ithaca NY 14853 USA

Intrigued by the product formula Pi(n-2)(i=1) C-i for the volume of the Chan-Robbins-Yuen polytope CRYn, where Ci is the i(th) Catalan number, we construct a family of polytopes P-m,P-n, indexed by m is an element of Z(>= 0) and n is an element of Z(>= 2), whose volumes are given by the product Pi(m+n-1)(i=m+1) 1/2i + 1((m + n + i + 1)(2i)) The Chan-Robbins-Yuen polytope CRYn coincides with P-0,P-n-2. Our construction of the polytopes P-m,P-n is an application of a systematic method we develop for expressing volumes of a class of flow polytopes as the number of certain triangular arrays. This method can also be used as a heuristic technique for constructing polytopes with combinatorial volumes. As an illustration of this we construct polytopes whose volumes equal the number of r-ary trees on n internal nodes, 1/(r-1)n+1((rn)(n)). Using triangular arrays we also express the volumes of flow polytopes as constant terms of formal Laurent series.

关键词： Flow polytope Chan-Robbins-Yuen polytope triangular arrays triangulation volume Kostant partition function

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