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作者：Westling, Ted; Carone, Marco
作者单位：University of Pennsylvania; University of Washington; University of Washington Seattle
摘要：The problem of nonparametric inference on a monotone function has been extensively studied in many particular cases. Estimators considered have often been of so-called Grenander type, being representable as the left derivative of the greatest convex minorant or least concave majorant of an estimator of a primitive function. In this paper, we provide general conditions for consistency and pointwise convergence in distribution of a class of generalized Grenander-type estimators of a monotone fun...
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作者：Candes, Emmanuel J.; Sur, Pragya
作者单位：Stanford University; Harvard University
摘要：This paper rigorously establishes that the existence of the maximum likelihood estimate (MLE) in high-dimensional logistic regression models with Gaussian covariates undergoes a sharp phase transition. We introduce an explicit boundary curve h(MLE), parameterized by two scalars measuring the overall magnitude of the unknown sequence of regression coefficients, with the following property: in the limit of large sample sizes n and number of features p proportioned in such a way that p/n -> kappa...
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作者：Ledoit, Olivier; Wolf, Michael
作者单位：University of Zurich
摘要：This paper establishes the first analytical formula for nonlinear shrinkage estimation of large-dimensional covariancematrices. We achieve this by identifying and mathematically exploiting a deep connection between nonlinear shrinkage and nonparametric estimation of the Hilbert transform of the sample spectral density. Previous nonlinear shrinkage methods were of numerical nature: QuEST requires numerical inversion of a complex equation from random matrix theory whereas NERCOME is based on a s...
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作者：Lopes, Miles E.; Lin, Zhenhua; Mueller, Hans-Georg
作者单位：University of California System; University of California Davis
摘要：In recent years, bootstrap methods have drawn attention for their ability to approximate the laws of max statistics in high-dimensional problems. A leading example of such a statistic is the coordinatewise maximum of a sample average of n random vectors in R-p. Existing results for this statistic show that the bootstrap can work when n << p, and rates of approximation (in Kolmogorov distance) have been obtained with only logarithmic dependence in p. Nevertheless, one of the challenging aspects...
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作者：Bunea, Florentina; Giraud, Christophe; Luo, Xi; Royer, Martin; Verzelen, Nicolas
作者单位：Cornell University; Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; University of Texas System; University of Texas Health Science Center Houston; University of Texas School Public Health; Universite de Montpellier; Institut Agro; Montpellier SupAgro; INRAE
摘要：The problem of variable clustering is that of estimating groups of similar components of a p-dimensional vector X = (X- 1, ..., X- p) from n independent copies of X. There exists a large number of algorithms that return data-dependent groups of variables, but their interpretation is limited to the algorithm that produced them. An alternative is model-based clustering, in which one begins by defining population level clusters relative to a model that embeds notions of similarity. Algorithms tai...
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作者：He, Yi; Hou, Yanxi; Peng, Liang; Shen, Haipeng
作者单位：University of Amsterdam; Fudan University; University System of Georgia; Georgia State University; University of Hong Kong
摘要：Conditional value-at-risk is a popular risk measure in risk management. We study the inference problem of conditional value-at-risk under a linear predictive regression model. We derive the asymptotic distribution of the least squares estimator for the conditional value-at-risk. Our results relax the model assumptions made in (Oper. Res. 60 (2012) 739-756) and correct their mistake in the asymptotic variance expression. We show that the asymptotic variance depends on the quantile density funct...
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作者：Shen, Yandi; Gao, Chao; Witten, Daniela; Han, Fang
作者单位：University of Washington; University of Washington Seattle; University of Chicago
摘要：Consider the heteroscedastic nonparametric regression model with random design Y-i = f (X-i) + V-1/2 (X-i)epsilon(i), i = 1, 2, ..., n, with f (.) and V (.) alpha- and beta-Holder smooth, respectively. We show that the minimax rate of estimating V (.) under both local and global squared risks is of the order n( - 8 alpha beta/4 alpha beta+2 alpha+beta )boolean OR n (- 2 beta/2 beta+1), where a boolean OR b := max{a, b} for any two real numbers a, b. This result extends the fixed design rate n(...
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作者：Gao, Chao; van der Vaart, Aad W.; Zhou, Harrison H.
作者单位：University of Chicago; Leiden University; Leiden University - Excl LUMC; Yale University
摘要：High dimensional statistics deals with the challenge of extracting structured information from complex model settings. Compared with a large number of frequentist methodologies, there are rather few theoretically optimal Bayes methods for high dimensional models. This paper provides a unified approach to both Bayes high dimensional statistics and Bayes nonparametrics in a general framework of structured linear models. With a proposed two-step prior, we prove a general oracle inequality for pos...
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作者：Ghoshdastidar, Debarghya; Gutzeit, Maurilio; Carpentier, Alexandra; von Luxburg, Ulrike
作者单位：Eberhard Karls University of Tubingen; Otto von Guericke University
摘要：The study of networks leads to a wide range of high-dimensional inference problems. In many practical applications, one needs to draw inference from one or few large sparse networks. The present paper studies hypothesis testing of graphs in this high-dimensional regime, where the goal is to test between two populations of inhomogeneous random graphs defined on the same set of n vertices. The size of each population m is much smaller than n, and can even be a constant as small as 1. The critica...
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作者：Kutyniok, Gitta
作者单位：Technical University of Berlin
摘要：I would like to congratulate Johannes Schmidt-Hieber on a very interesting paper in which he considers regression functions belonging to the class of so-called compositional functions and analyzes the ability of estimators based on the multivariate nonparametric regression model of deep neural networks to achieve minimax rates of convergence. In my discussion, I will first regard such a type of result from the general viewpoint of the theoretical foundations of deep neural networks. This will ...
