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

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2 Author(s)

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) = -\int f(x) \ln f(x) dx . In this correspondence we propose, based on a random sample X_{1}, \cdots , X_{n} generated from F , a nonparametric estimate of H(f) given by \hat{H}(f) = -(l/n) \sum _{i = 1}^{n} In \hat{f}(x) , where \hat{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 \hat{H}(f) are established. These conditions are mild and easily satisfied. Examples, such as Gamma, Weibull, and normal distributions, are considered.

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Information Theory, IEEE Transactions on  (Volume:22 ,  Issue: 3 )