Geodesic
On estimation of functionals of the probability density function and its derivatives
Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 3, pp. 662-668

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For functionals of the type $I=\int_{-\infty}^\infty H(f(y)),f'(y),\dots,f^{(r)}(y))\,dy$ the estimates $I_N=\int_{-k_N}^{k_N}H(f_N(y),\dots,f_N^{(r)}(y))\,dy$ are considered. Here $f_N(y),\dots,f_N^{(r)}(y)$ are nonparametric estimates of the density and of its derivatives introduced by Rosenblatt and studied by Parzen, Bhattacharya, Nadaraya and others. Theorems on convergence of the estimates with probability one are proved for Fisher's information, the entropy and the integral of the squared density. Convergence in probability are also investigated. 
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     author = {Yu. G. Dmitriev and F. P. Tarasenko},
     title = {On estimation of functionals of the probability density function and its derivatives},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     volume = {18},
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     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1973_18_3_a26/}
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Yu. G. Dmitriev; F. P. Tarasenko. On estimation of functionals of the probability density function and its derivatives. Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 3, pp. 662-668. http://geodesic.mathdoc.fr/item/TVP_1973_18_3_a26/