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Teoriya Veroyatnostei i ee Primeneniya, 1973, Volume 18, Issue 3, Pages 662–668
(Mi tvp2747)
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This article is cited in 50 scientific papers (total in 50 papers)
Short Communications
On estimation of functionals of the probability density function and its derivatives
Yu. G. Dmitriev, F. P. Tarasenko Tomsk
Abstract:
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.
Received: 20.12.1971
Citation:
Yu. G. Dmitriev, F. P. Tarasenko, “On estimation of functionals of the probability density function and its derivatives”, Teor. Veroyatnost. i Primenen., 18:3 (1973), 662–668; Theory Probab. Appl., 18:3 (1974), 628–633
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