ON THE ESTIMATION OF ENTROPY

PETER HALL^{1} AND SALLY C. MORTON^{2}

1Centre for Mathematics and its Applications, Australian National University, G.P.O. Box 4, Canberra A.C.T. 2601, Australia and CSIRO Division of Mathematics and Statistics 2Statistical Research and Consulting Group, The RAND Corporation, 1700 Main Street, P.O. Box 2138, Santa Monica, CA 90407-2138, U.S.A.

(Received May 7, 1991; revised April 14, 1992)

Abstract. Motivated by recent work of Joe (1989, Ann. Inst. Statist. Math., 41, 683–697), we introduce estimators of entropy and describe their properties. 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 assumptions about each of these three features. Our estimators are different from Joe’s, and may be computed without numerical integration, but it can be shown that the same interaction of tail behaviour, smoothness and dimensionality also determines the convergence rate of Joe’s estimator. We study both histogram and kernel estimators of entropy, and in each case suggest empirical methods for choosing the smoothing parameter.

Key words and phrases: Convergence rates, density estimation, entropy, histogram estimator, kernel estimator, projection pursuit, root-n consistency.