Nonparametric functional estimation.

*(English)*Zbl 0542.62025
Probability and Mathematical Statistics. Orlando, Florida, etc.: Academic Press (Harcourt Brace Jovanovich, Publishers). XIV, 522 p. $ 70.00 (1983).

This book contains a long chain of various methods on nonparametric estimation of density functions, distribution functions, regression functions, failure rates, etc., gathered from a vast literature (the list of references comprises nearly thirty pages!).

The presentation is fairly theoretical being of use more for the specialist in this area than for students who might miss necessary motivation and some more links between different subjects which are organized as follows:

1. Estimation of functionals. 2. Density estimation (univariate case). 3. Density estimation (multivariate case). 4. Estimation of functionals related to density. 5. Sequential and recursive estimation. 6. Estimation for stochastic processes. 7. Estimation under order restrictions. 8. Nonparametric discrimination. 9. Estimation of a distribution function. 10. Estimation of a mixing distribution. 11. Bayes estimation.

As the author mentions in the preface, the main emphasis throughout the book is on the discussion of several methods of estimation and on the study of their large sample properties. This is done in a rather thorough way. Many results (some of which are presented as exercises at the end of each chapter together with an appropriate reference) appear here in book form for the first time. In view of the high price to be paid for this book one could have had expected a more careful proof reading (cf. e.g. the references to H. Bauer (1968) on p. 485 and to W. Wertz (1972b and 1972c) on p. 511).

The presentation is fairly theoretical being of use more for the specialist in this area than for students who might miss necessary motivation and some more links between different subjects which are organized as follows:

1. Estimation of functionals. 2. Density estimation (univariate case). 3. Density estimation (multivariate case). 4. Estimation of functionals related to density. 5. Sequential and recursive estimation. 6. Estimation for stochastic processes. 7. Estimation under order restrictions. 8. Nonparametric discrimination. 9. Estimation of a distribution function. 10. Estimation of a mixing distribution. 11. Bayes estimation.

As the author mentions in the preface, the main emphasis throughout the book is on the discussion of several methods of estimation and on the study of their large sample properties. This is done in a rather thorough way. Many results (some of which are presented as exercises at the end of each chapter together with an appropriate reference) appear here in book form for the first time. In view of the high price to be paid for this book one could have had expected a more careful proof reading (cf. e.g. the references to H. Bauer (1968) on p. 485 and to W. Wertz (1972b and 1972c) on p. 511).

Reviewer: P.Gänßler

##### MSC:

62G05 | Nonparametric estimation |

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

62M05 | Markov processes: estimation; hidden Markov models |

62H30 | Classification and discrimination; cluster analysis (statistical aspects) |

62L12 | Sequential estimation |

62M09 | Non-Markovian processes: estimation |