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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 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)
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)
This paper develops a uniform test of linearity against thresholds effects in the quantile regression framework. The test is based on the supremum of the Wald process over the space of quantile and threshold parameters. We establish the asymptotic null distribution of the test statistic for stationary weakly dependent processes, and propose a simulation(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)
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 paper studies panel quantile regression models with fixed effects. We formally establish sufficient conditions for consistency and asymptotic normality of the quantile regression estimator when the number of individuals, n, and the number of time periods, T , jointly go to infinity. The estimator is shown to be consistent under similar conditions to(More)