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- W2014560315 abstract "Recently, attention has been drawn to the problem of estimation of a k -variate probability density and its partial derivatives of various orders. Specifically, let X 1 , …, X n be i.i.d. k -variate random variables with common density f wrt Lebesgue measure μ on the k -dimensional σ-field B k . Parzen (1962) in the k = 1 case and Cacoullos (1966) in the k ≧ 1 case gave the asymptotic properties of a class of kernel estimates f n ( x ), x ∈ R k , of f(x) based on X 1 , …, X n . The asymptotic properties given in the above two papers concern consistency, asymp-totic unbiasedness, bounds for the mean squared error and asymptotic normality of f n . Also in the context of an empirical Bayes two-action problem, Johns and Van Ryzin (1972) introduced kernel estimates for f (x) and the derivative f '(x)for x∈ R 1 when f is a mixture of univariate exponential densities wrt Lebesgue measure on B 1 . They also investigated the asymptotic unbiasedness and themean squared error convergence properties of these estimates. Lin (1968) statedsome generalizations of the results of Johns and Van Ryzin, with applicationsto empirical Bayes decision problems." @default.
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- W2014560315 title "Estimation of Partial derivatives of the average of densities belonging to a family of densities" @default.
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