A nonparametric estimation of the entropy for absolutely continuous distributions (Corresp.)

Let F (x) be an absolutely continuous distribution having a density function f (x) with respect to the Lebesgue measure. The Shannon entropy is defined as H (f) = - f (x) f (x) dx. In this correspondence we propose, based on a random sample X₁, , X₍ generated from F, a nonparametric estimate of H (f) given by H (f) = - (l/n) ₈ = ₁^n f (x), where f (x) is the kernel estimate of f due to Rosenblatt and Parzen. Regularity conditions are obtained under which the first and second mean consistencies of H (f) are established. These conditions are mild and easily satisfied. Examples, such as Gamma, Weibull, and normal distributions, are considered.