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We derive a Gaussian approximation result 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. This result applies when the(More)
Slepian and Sudakov-Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution(More)
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,(More)
We study the degrees of freedom in shrinkage estimation of the regression coefficients. Generalizing the idea of the Lasso, we consider the problem of estimating the coefficients by the projection of the ordinary least squares estimator onto a closed convex set. Then an unbiased estimator of the degrees of freedom is derived in terms of geometric quantities(More)
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Modern construction of uniform confidence bands for nonpara-metric densities(More)
This work proposes new inference methods for the estimation of a regression coefficient of interest in quantile regression models. We consider high-dimensional models where the number of regressors potentially exceeds the sample size but a subset of them suffice to construct a reasonable approximation of the unknown quantile regression function in the(More)
Due to the morphological variability, the identification of moss species can be difficult when the plant grows in submerged environments. The taxonomic status of an aquatic moss found in lakes of the Sôya Coast region, East Antarctica, had been controversial, and then, it was investigated by molecular phylogenetic and haplotype network analysis of two(More)