The Weighted Bootstrap. Lecture Notes in Statistics 98

Barbe, Philippe, Bertail, Patrice

ISBN 10: 0387944788 / ISBN 13: 9780387944784

Editore: Springer, 1995

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Codice articolo ZB1332765

Dati bibliografici

• Titolo: The Weighted Bootstrap. Lecture Notes in Statistics 98
• Editore: Springer
• Data di pubblicazione: 1995
• Lingua: Inglese
• ISBN 10: 0387944788
• ISBN 13: 9780387944784
• Rilegatura: Brossura
• Condizione: Fine

Descrizione articolo

Riassunto

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.

Contenuti

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) Fréchet-differentiability and metric indexed by a class of functions.- I.2.1) Differentiability assumptions.- I.2.2) The choice of the metric.- I.2.3) Rate of convergence of the weighted empirical process indexed by a class of functions.- I.3) Consistency of the generalized bootstrapped distribution, variance estimation and Edgeworth expansion.- I.3.1) Consistency of the generalized bootstrapped distribution.- I.3.2) The generalized bootstrap variance estimator.- I.3.3) Edgeworth expansion of the studentized functional.- I.3.4) Inverting Edgeworth expansion to construct confidence intervals.- I.4) Applications.- I.4.1) The mean.- I.4.2) M-estimators.- I.4.3) The probability of being censored.- I.4.4) Multivariate V-statistics.- 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.3.1) Second order correction.- II.3.2) Coverage probability.- II.4) Choice of the weights for general functional via Edgeworth expansion.- II.4.1) Edgeworth expansion up to o(n-1) for a third order differentiable functional.- II.4.2) Edgeworth Expansion up to o(n-1) for the weighted version.- II.5) Coverage probability for the weighted bootstrap of general functional.- II.5.1) Derivation of the coverage probability.- II.5.2) Choosing the weights via minimization of the coverage probability.- II.5.3) Simulation results.- 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 Theorem III.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.