zbMATH — the first resource for mathematics

The bootstrap and Edgeworth expansion. (English) Zbl 0744.62026
Springer Series in Statistics. New York etc.: Springer-Verlag. xiii, 352 p. (1992).
This monograph deals with two topics, the theory of Edgeworth expansion and the theory of bootstrap, due to Efron, with an attempt to relate the two theories. It is emphasized that methods based on Edgeworth expansion explain the performance of bootstrap methods, whereas the bootstrap provides a strong motivation for examining the theory of Edgeworth expansion. The study of bootstrap is largely confined to a collection of problems which can be treated using the classical theory of sums of independent random variables. The study is directed mainly at problems of estimating distributions, wherein the theory of Edgeworth expansion is most relevant.
The material is covered in five chapters and five appendices. The bootstrap and Edgeworth expansion are discussed in Chapters 1 and 2, respectively. Chapter 3 brings the two themes together. Chapter 4 continues the approach along less conventional lines and Chapter 5 sketches details of mathematical rigour that are missing from earlier chapters. The appendices complement the theoretical themes of the monograph.
Reviewer: K.Alam (Clemson)

62E20 Asymptotic distribution theory in statistics
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62G09 Nonparametric statistical resampling methods
60G40 Stopping times; optimal stopping problems; gambling theory