Ebook: The Weighted Bootstrap

INTRODUCTION 1) Introduction In 1979, Efron introduced the bootstrap method as a kind of universal tool to obtain approximation of the distribution of statistics. The now well known underlying idea is the following : consider a sample X of Xl ' n independent and identically distributed H.i.d.) random variables (r. v,'s) with unknown probability measure (p.m.) P . Assume we are interested in approximating the distribution of a statistical functional T(P ) the -1 nn empirical counterpart of the functional T(P) , where P n := n l:i=l aX. is 1 the empirical p.m. Since in some sense P is close to P when n is large, n • • LLd. from P and builds the empirical p.m. if one samples Xl ' ... , Xm n n -1 mn • • P T(P ) conditionally on := mn l: i =1 a • ' then the behaviour of P m n,m n n n X. 1 T(P ) should imitate that of when n and mn get large. n This idea has lead to considerable investigations to see when it is correct, and when it is not. When it is not, one looks if there is any way to adapt it.

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This monograph presents an account of the asymptotic behaviour of the weighted bootstrap - a new and powerful statistical technique. Researchers and advanced graduate students studying bootstrap methods will find this a valuable technical survey which is thorough and rigorous.
The main aim of this book is to answer two questions: How well does the generalized bootstrap work? What are the differences between all the different weighted schemes? Readers are assumed to have already some familiarity with the bootstrap, but otherwise the account is as self-contained as possible. Proofs are presented in detail, though some lengthy calculations are deferred to appendices.

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This monograph presents an account of the asymptotic behaviour of the weighted bootstrap - a new and powerful statistical technique. Researchers and advanced graduate students studying bootstrap methods will find this a valuable technical survey which is thorough and rigorous.
The main aim of this book is to answer two questions: How well does the generalized bootstrap work? What are the differences between all the different weighted schemes? Readers are assumed to have already some familiarity with the bootstrap, but otherwise the account is as self-contained as possible. Proofs are presented in detail, though some lengthy calculations are deferred to appendices.
Content:
Front Matter....Pages ii-x
Introduction....Pages 1-8
Asymptotic Theory for the Generalized Bootstrap of Statistical Differentiable Functionals....Pages 9-43
How to Choose the Weights....Pages 45-76
Special Forms of the Bootstrap....Pages 77-91
Proofs of Results of Chapter I....Pages 93-118
Proofs of Results of Chapter II....Pages 119-144
Proofs of Results of Chapter III....Pages 145-152
Back Matter....Pages 153-233

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This monograph presents an account of the asymptotic behaviour of the weighted bootstrap - a new and powerful statistical technique. Researchers and advanced graduate students studying bootstrap methods will find this a valuable technical survey which is thorough and rigorous.
The main aim of this book is to answer two questions: How well does the generalized bootstrap work? What are the differences between all the different weighted schemes? Readers are assumed to have already some familiarity with the bootstrap, but otherwise the account is as self-contained as possible. Proofs are presented in detail, though some lengthy calculations are deferred to appendices.
Content:
Front Matter....Pages ii-x
Introduction....Pages 1-8
Asymptotic Theory for the Generalized Bootstrap of Statistical Differentiable Functionals....Pages 9-43
How to Choose the Weights....Pages 45-76
Special Forms of the Bootstrap....Pages 77-91
Proofs of Results of Chapter I....Pages 93-118
Proofs of Results of Chapter II....Pages 119-144
Proofs of Results of Chapter III....Pages 145-152
Back Matter....Pages 153-233
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