Approximate Distributions of Order Statistics: With Applications to Nonparametric StatisticsThis 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) that are relevant for statistical theory and its applications. Particular emphasis is placed on results concern ing the accuracy oflimit 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 approxi mating models include normal as well as extreme value models. Several applications will show the usefulness of this approach. Other recent developments in statistics like nonparametric curve estima tion and the bootstrap method will be studied as far as order statistics are concerned. 1n connection with this, graphical methods will, to some extent, be explored. |
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Approximate Distributions of Order Statistics: With Applications to ... Rolf-Dieter Reiss Limited preview - 2012 |
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Common terms and phrases
applied Assume asymptotic normality b₁ bootstrap Borel sets br,n central order statistics common d.f. conditional distribution constant continuous d.f. Corollary defined Denote density f distributions of order domain of attraction estimator Example expansion of length extreme value d.f. extreme value theory F₁ F¹(q function given Gumbel Hellinger distance holds i.i.d. random variables implies inequality joint density joint distribution Kullback-Leibler distance largest order statistics Lemma limiting d.f.'s Markov kernel Moreover multivariate N₁ normal distribution Notice obtain Pareto d.f. polynomials positive integer probability measure prove q-quantile Q₁ r₁ random vectors Reiss rth order statistic S₁ sample maxima sample median sample q.f. sample quantiles Section signed measures standard normal underlying d.f. uniform distribution uniformly unimodal upper bound variables with common variational distance weak convergence Weibull densities x₁ Xn:n Xs:n y₁ zero