Approximate distributions of order statistics. With applications to nonparametric statistics.

*(English)*Zbl 0682.62009
Springer Series in Statistics. New York etc.: Springer-Verlag. xii 355 p. DM 124.00 (1989).

The probability distributions of order statistics can be derived by elementary techniques, but are usually too complicated for convenient use. As a result, much research has been devoted to constructing approximate distributions for order statistics, but until very recently very little information was available about how good these approximate distributions are.

The present monograph supplies expansions for the distributions of both central and extreme order statistics, and error bounds for the approximations. Many of the results are due to the author of the monograph. The section on applications to nonparametric statistics gives estimates of the quantile function and the density quantile function corresponding to an unknown distribution, and discusses the asymptotic sufficiency of a relatively small number of order statistics for statistical inference. The bibliography contains about 400 entries, which gives some indication of the scope of this monograph. There are also problems for the reader, so it would serve as an advanced text.

The present monograph supplies expansions for the distributions of both central and extreme order statistics, and error bounds for the approximations. Many of the results are due to the author of the monograph. The section on applications to nonparametric statistics gives estimates of the quantile function and the density quantile function corresponding to an unknown distribution, and discusses the asymptotic sufficiency of a relatively small number of order statistics for statistical inference. The bibliography contains about 400 entries, which gives some indication of the scope of this monograph. There are also problems for the reader, so it would serve as an advanced text.

Reviewer: L.Weiss

##### MSC:

62E20 | Asymptotic distribution theory in statistics |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62G30 | Order statistics; empirical distribution functions |

62E15 | Exact distribution theory in statistics |