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A course in density estimation. (English) Zbl 0617.62043
Progress in Probability and Statistics, Vol. 14. Boston/Basel/Stuttgart: Birkhäuser. XIX, 183 p.; DM 58.00 (1987).
This book has been written from a course taught during summer 1986 at Stanford University. It is organized like lecture notes for graduate students and can be seen as a didactic introduction to research monographs in density estimation: it completes very well the book written by the present author jointly with L. Györfi [Nonparametric density estimation: The \(L_ 1\) view. (1985; Zbl 0546.62015)] which must be reserved to people already acquainted with functional estimation.
The subject of the book is in fact more specialized than its title says: it deals almost entirely with kernel estimates and therefore the beginner will have only a partial view on density estimation. For instance nothing is said about the orthogonal functions method or the non iid case. Density estimation is, nowadays, an extensive matter and the present text is, therefore, the result of a choice, perfectly argued for by the author in its preface.
The first chapter deals with distances between densities and Kullback- Leibler information; fundamental inequalities are given. The second one presents the key ideas which are developed in the following and the third one gives \(L_ 1\) consistency results about the kernel estimate.
Considerations about bias reduction and rates of convergence will be found in chapter seven. Robustness is examined in chapter four. The criterion selected to define this notion is E(\(\int | f_ n- f|)\). Chapter five is devoted to the computation of minimax bounds. The parts played respectively by the smoothness and the tail of the density are investigated. Minimum distance estimators are built in chapter six.
Chapter eight is a case study: monotone densities on [0,1]. Various estimates are compared. One may regret that monotone splines are not mentioned here. Finally chapter nine deals with relative stability in terms of \(L_ 1\) criterion.
All along the book more than fifty figures illustrate the text and about the same number of well adapted exercises authorize further study. The care given to the preface (ten pages) must be noticed: the main ideas of the book are pointed out and a number of application domains as well as open fields in density estimation are shown. This book integrates a number of recent works and explains the difficulties even when they cannot be mathematically detailed at such a level. It is pleasant by its typographic aspect and attractive by its scientific contents. I highly recommend it as a basic tool to all people coming to density estimation.
Reviewer: A.Berlinet

MSC:
62G05 Nonparametric estimation
62-02 Research exposition (monographs, survey articles) pertaining to statistics