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Weighted uniform consistency of kernel density estimators

2004
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Annals of Probability
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Let f_n denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let \Psi(t) be a positive continuous function such that \|\Psi f^{\beta}\|_{\infty}<\infty for some 0<\beta<1/2. Under natural smoothness conditions, necessary and sufficient conditions for the sequence \sqrt\frac{nh_n^d}{2|\log h_n^d|}\|\Psi(t)(f_n(t)-Ef_n(t))\|_{\infty} to be stochastically bounded and to converge a.s. to a constant are obtained. Also, the case of larger values of \beta

doi:10.1214/009117904000000063
fatcat:x57jbl2mibd3jfspyeae6u4uoy