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.
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References
Friedman, J. H. (1987). Exploratory projection pursuit,J. Amer. Statist. Assoc.,76, 817–823.
Goldstein, L. and Messer, K. (1991). Optimal plug-in estimators for nonparametric functional estimation,Ann. Statist. (to appear).
Hall, P. (1987). On Kullback-Leibler loss and density estimation,Ann. Statist.,15, 1491–1519.
Hall, P. (1989). On polynomial-based projection indices for exploratory projection pursuit,Ann. Statist.,17, 589–605.
Hall, P. (1990). Akaike's information criterion and Kullback-Leibler loss for histogram density estimation,Probab. Theory Related Fields,85, 449–466.
Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution,Ann. Statist.,3, 1163–1174.
Huber, P. J. (1985). Projection pursuit (with discussion),Ann. Statist.,13, 435–525.
Joe, H. (1989). Estimation of entropy and other functionals of a multivariate density,Ann. Inst. Statist. Math.,41, 683–697.
Jones, M. C. and Sibson, R. (1987). What is projection pursuit? (with discussion),J. Roy. Statist. Soc. Ser. B,150, 1–36.
Morton, S. C. (1989). Interpretable projection pursuit, Ph.D. Dissertation, Stanford University, California.
Silverman, B. W. (1986).Density Estimation for Statistics and Data Analysis, Chapman and Hall, London.
Vasicek, O. (1986). A test for normality based on sample entropy,J. Roy. Statist. Soc. Ser. B,38, 54–59.
Zipf, G. K. (1965).Human Behaviour and the Principle of Least Effort: An Introduction to Human Ecology (Facsimile of 1949 edn.). Hafner, New York.
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Hall, P., Morton, S.C. On the estimation of entropy. Ann Inst Stat Math 45, 69–88 (1993). https://doi.org/10.1007/BF00773669
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DOI: https://doi.org/10.1007/BF00773669