The Weighted Bootstrap - Patrice Bertail

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Résumé :
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.

Sommaire:
Table.- I.1) Introduction.- I.2) Some connected works.- I) Asymptotic theory for the generalized bootstrap of statistical differentiate functionals.- I.1) Introduction.- I.2) Fr?chet-differentiability and metric indexed by a class of functions.- I.3) Consistency of the generalized bootstrapped distribution, variance estimation and Edgeworth expansion.- I.4) Applications.- I.5) Some simulation results.- II) How to choose the weights.- II.1) Introduction.- II.2) Weights generated from an i.i.d. sequence : almost sure results.- II.3) Best weights for the bootstrap of the mean via Edgeworth expansion.- II.4) Choice of the weights for general functional via Edgeworth expansion.- II.5) Coverage probability for the weighted bootstrap of general functional.- II.6) Conditional large deviations.- II.7) Conclusion.- III) Some special forms of the weighted bootstrap.- III.1) Introduction.- III.2) Bootstrapping an empirical d.f. when parameters are estimated or under some local alternatives.- III.3) Bootstrap of the extremes and bootstrap of the mean in the infinite variance case.- III.4) Conclusion.- IV) Proofs of results of Chapter I.- IV.1) Proof of Proposition I.2.1.- IV.2) Proof of Proposition I.2.2.- IV.3) Proof of Theorem I.3.1.- IV.4) Some notations and auxilliary lemmas.- IV.5) Proof of Theorem I.3.2.- IV.6) More lemmas to prove Theorem I.3.2.- IV.7) Proof of Theorem I.3.3.- IV.8) Proof of Theorem I.3.4.- IV.9) Proof of Theorem I.3.5.- V) Proofs of results of Chapter II.- V.1) Proofs of results of section II. 2.- V.2) Proof of Formula (II.3.2).- V.3) Proof of Proposition II.4.1.- V.4) Proof of (II.5.6).- V.5) Proof of (II.5.9).- V.6) Proof of (II.5.10).- V.7) Proof of (II.5.11).- V.8) Proof of Theorem II.6.2.- VI) Proofs of results of Chapter III.- VI.1) Proof of TheoremIII.1.1.- VI.2) Proof of Theorem III.1.2.- VI.3) Proof of Theorem III.2.1.- VI.4) Proof of Theorem III.2.2.- Appendix 1 : Exchangeable variables of sum 1.- Appendix 5 : Finite sample asymptotic for the mean and the bootstrap mean estimator.- Appendix 6 : Weights giving an almost surely consistent bootstrapped mean.- References.- Notation index.- Author index.