On the asymptotic theory of the bootstrap.
Item
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Title
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On the asymptotic theory of the bootstrap.
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Identifier
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AAI9130290
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identifier
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9130290
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Creator
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Arcones, Miguel Angel.
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Contributor
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Adviser: Evarist Gine
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Date
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1991
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics | Statistics
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Abstract
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This dissertation is a compilation of five papers by the author of this thesis and his advisor about the bootstrap in different situations.;The first chapter is an introduction to the topic.;The second chapter continues the work of Gine and Zinn (1990) and Athreya (1987) on the bootstrap of the mean in the absence of second moment and arbitrary sample size {dollar}m\sb{lcub}n{rcub}{dollar}. Here we show among other results, that (1) if X is in the domain of attraction of a normal random variable the bootstrap always holds in probability and that (2) if X is in a domain of attraction of a stable law and {dollar}EX\sp2{dollar} = {dollar}\infty{dollar}, then bootstrap CLT holds a.s. if {dollar}m\sb{lcub}n{rcub}{dollar} log log {dollar}n/n{dollar} {dollar}\to{dollar} 0 and does not hold a.s. if sup {dollar}m\sb{lcub}n{rcub}{dollar} log log {dollar}n/n{dollar} {dollar}>{dollar} 0. (3) the convergence of bootstrap moments.;In Chapter III we examine the bootstrap of U-statistics particularly in the degenerate case. In this case, Bretagnolle showed that the naive bootstrap also holds a.s. for U-statistics under some additional conditions on the moments if {dollar}m\sb{lcub}n{rcub}{dollar}/log {dollar}n{dollar} {dollar}\to{dollar} 0. Here we present a modified bootstrap CLT that holds a.s. for any {dollar}m\sb{lcub}n{rcub}{dollar} in particular for {dollar}m\sb{lcub}n{rcub}{dollar} = {dollar}n{dollar} under weak integrabiblity conditions. The modification consists in bootstrapping the first non-null term of the Hoeffding expansion with respect to {dollar}P\sb{lcub}n{rcub}{dollar}.;Chapter IV deals with the bootstrap of a test of symmetry. We give mathematical justification to the method used in Barker and Schuster (1988). Moreover we give 3 other different tests.;Finally Chapter V contains the bootstrap of some M-estimators whose asymptotics can be deduced from the theory of empirical measures, through the results about the bootstrap of empirical measures in Gine and Zinn (1990). The M-estimators we consider are in the framework of Huber (1967) and Pollard (1985). They cover most M-estimators used in practice. We obtain, as examples of the general theory, the a.s. bootstrap of the spatial medians, k-means and Huber estimators.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
