Approximate distributions of order statistics: with applications to nonparametric statistics
This book is designed as a unified and mathematically rigorous treatment of some recent developments of the asymptotic distribution theory of order statistics (including the extreme order statistics) which are relevant for statistical theory and its applications. Particular emphasis is placed on results concerning the accuracy of limit theorems, on higher order approximations and other approximations in quite a general sense. Contrary to the classical limit theorems, that primarily concern the weak convergence of distribution functions, our main results will be formulated in terms of the variational and the Hellinger distance. These results will form the proper springboard for the investigation of parametric approximations of nonparametric models of joint distributions of order statistics. The approximating models include normal as well as extreme value models. Several applications will show the usefulness of this approach. This book is intended for students and research workers in probability and statistics, and practitioners involved in applications of mathematical results conderning order statistics and extremes. The knowledge of calculus and topics that are taught in introductory probability and statistic courses are necessary for the understanding of this book.
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absolutely continuous applied Assume asymptotic normality bootstrap Borel sets central order statistics co(F common d.f. conditional distribution Corollary defined Denote derivatives distributions of order domain of attraction estimator Example expansion of length extreme value d.f. extreme value theory function g-quantile given Hellinger distance holds i.i.d. random variables implies inequality joint density joint distribution largest order statistics Lemma Let Xi:n limiting d.f.'s Markov kernel maximum Mises Moreover nondecreasing normal distribution normalizing constants Notice obtain Pareto d.f. polynomials positive integer probability measure proof of Theorem prove random vectors Reiss rth order statistic sample maxima sample median sample q.f. sample quantiles Section signed measures single order statistic standard exponential r.v.'s standard normal underlying d.f. uniform distribution uniform r.v.'s uniformly unimodal upper bound Ur:n variables with common variational distance w.r.t. the variational weak convergence Weibull densities Xn:n Xs:n yields zero