This book is based on notes used in a summer quarter course given at Stanford in 1986. It consists of nine chapters, each concluding with a set of exercises. The preface is must reading as it succinctly summarizes the text and serves as an excellent guide to its contents.
Chapter 1 is entitled “Distance Between Densities” and establishes the relationships between the Lp distance, the Hellinger distance, and the Kullback-Leibler number. The relationships are introduced in terms of seven theorems that make up the bulk of the chapter. Each theorem is preceded by a motivating discussion.
The basic problem of density estimation is introduced in chapter 2. Several nonparametric and tailor-designed density estimates are derived here; the main thread is to estimate the kernel. Chapter 3 is devoted to illustrating some techniques for proving the consistency of nonparametric estimates, using estimates of the kernel as the main example.
In chapter 4, robustness of density estimates in terms of L1 distances is defined. The main point here is to show that robustness is equivalent to insensitivity to small changes in the sample.
Chapter 5 deals with lower bounds on the expected error involved in density estimation using information-theoretic methods. A systematic method for constructing minimax-optimal estimates is presented in chapter 6. This is based on minimum density estimates.
Chapter 7 deals with the relationship between the smoothness of a density and the best possible rates of convergence that can be attained by the kernel estimate. In chapter 8 a case study on monotone density estimation is presented and various estimates for this problem are compared. Finally, chapter 9 deals with the issue of relative stability, which is crucial in determining the error criterion.
The text material is abstract and will appeal to a reader with an advanced grasp of statistics and related mathematics.
