ON THE ESTIMATION OF ENTROPY

Motivated by recent work of Joe (1989, Ann. Inst. Statist. Math., 41, 683-697), we introduce estimators of entropy and describe their proper ties. We study the effects of tail behaviour, distribution smoothness and dimensionality on convergence properties. In particular, we argue that root-n consistency of entropy estimation requires appropriate ass umptions about each of these three features. Our estimators are differ ent from Joe's, and may be computed without numerical integration, but it can be shown that the same interaction of tail behaviour, smoothne ss and dimensionality also determines the convergence rate of Joe's es timator. We study both histogram and kernel estimators of entropy, and in each case suggest empirical methods for choosing the smoothing par ameter.