Monographs on Statistics and Applied Probability. London - New York: Chapman and Hall. IX, 175 p.; £ 12.00 (1986).

In a review [Ann. Stat. 13, 1630-1638 (1985)] of the books ”Nonparametric functional estimation” (1983;

Zbl 0542.62025) by this reviewer and ”Nonparametric density estimation: The $L\sb 1$ view” (1985;

Zbl 0546.62015) by {\it L. Devroye} and {\it L. Györfi}, the present author has indicated that”... In order to balance the view that might be obtained from reading only the two books under review and nothing else, it may be worth briefly discussing a few practical contexts in which density estimators arise...”. He has done so in his book under review.
The reviewer’s book deals with nonparametric estimation of distribution function, regression function, failure rate etc. besides estimation of density function. The present author discusses various aspects of density estimation and its application through examples from real life data. Any scientific theory or method should be based on solid theoretical foundation buttressed with useful and interesting applications. Both aspects should go hand in hand. Same is the case with nonparametric density estimation. While it is true that much research work is done in studying large sample properties of various types of density estimators, applications of density estimation to practical problems is relatively meager at this time and there is no monograph dealing with this aspect of the problem. The present book fills this gap.
After a brief explanation for density estimation in practice in Chapter 1, the author gives a survey of existing methods in Chapter 2. Several aspects of the kernel method for density estimation for univariate and multivariate data are given in Chapters 3 and 4. Properties of the method of nearest neighbors, adaptive kernel estimation and maximum penalized likelihood estimators for density estimation are investigated in Chapter 5. Methods of choosing kernel and band width for kernel type estimators, computational considerations, and problems of density estimation in higher dimensional spaces are discussed in Chapter 4. Study of asymptotic properties of the estimators received little attention throughout the book as has been the author’s aim. Chapter 6 deals with density estimation in action. Applications to nonparametric discriminant analysis, cluster analysis, bump hunting and testing for multimodality are given in this chapter.
The book is written in a leisurely style and it is an excellent addition to the literature on density estimation. Any student or practitioner of density estimation can now hope to get a balanced view of both the theoretical and practical aspects of the problem by studying not only the books listed at the beginning of this review but also the book by the author.