The Weighted Bootstrap

By: Philippe Barbe, Patrice Bertail

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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 â¢ â¢ LLd. from P and builds the empirical p.m. if one samples Xl ' ... , Xm n n -1 mn â¢ â¢ P T(P ) conditionally on := mn l: i =1 a â¢ ' 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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Acknowledgements
Introduction	p. 1
Asymptotic theory for the generalized bootstrap of statistical differentiable functionals	p. 9
How to choose the weights	p. 45
Some special forms of the weighted bootstrap	p. 77
Proofs of results of Chapter I	p. 93
Proofs of results of Chapter II	p. 119
Proofs of results of Chapter III	p. 145
Appendix 1: Exchangeable variables of sum 1	p. 153
Appendix 2: How to express [actual symbol not reproducible] as a sum of U-statistics	p. 159
Appendix 3: Derivation of [actual symbol not reproducible]	p. 163
Appendix 4: Calculation of E[subscript W] Q[subscript W,n]	p. 169
Appendix 5: Finite sample asymptotic for the mean and the bootstrap mean estimator	p. 175
Appendix 6: Weights giving an almost surely consistent bootstrapped mean	p. 177
Appendix 7: Differentials of the functional T[superscript (1)](X,P)	p. 181
Appendix 8: Edgeworth expansions up to o(n[superscript -1]) for a third order differentiable statistical functional and its studentized form	p. 183
References	p. 199
Notation index	p. 215
Author index	p. 223
Subject index	p. 229
Table of Contents provided by Blackwell. All Rights Reserved.

The Weighted Bootstrap

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