2.11. Discussion and bibliography

A brief survey of the kind conducted in this chapter of course asks far more questions than it answers, and some of these questions will be the subject of discussion in subsequent chapters. The overriding problems are the choice of what method to use in any given practical context and, given that a particular method is being used, how to choose the various parameters needed by the method. The remarks already made about the mathematical properties of the estimates obtained by various procedures will of course be important in making these decisions. To obtain a fuller understanding of the importance and consequences of the various choices it is essential to investigate the statistical properties of the various methods and also to consider the difficulties involved in computing the estimates.

This chapter has by no means considered all the methods available for density estimation. Generalizations and other approaches are considered in later chapters of this book, and in the other books and surveys mentioned in Section 1.3.

The naive estimator was introduced by Fix and Hodges (1951) in an unpublished report; the first published paper to deal explicitly with probability density estimation was by Rosenblatt (1956), who discussed both the naive estimator and the more general kernel estimator. Whittle (1958) formulated the general weight function class of estimators, while the orthogonal series estimator was introduced by Cencov (1962). The nearest neighbour estimate was first considered by Loftsgaarden and Quesenberry (1965), while the variable kernel method is due to Breiman, Meisel and Purcell (1977), though Wertz (1978, p. 59) refers to presumably independent but related work by Victor. The maximum penalized likelihood approach was first applied to density estimation by Good and Gaskins (1971). The reflection and replication techniques of Section 2.10 were introduced and illustrated by Boneva, Kendall and Stefanov (1971), while the transformation technique is discussed by Copas and Fryer (1980).