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Minimax kernels for density estimation with biased data. (English) Zbl 0926.62027

Summary: This paper considers the asymptotic properties of two kernel estimates \(\widetilde{f}_n\) and \(\widehat{f}_n\) which have been proposed by B. B. Bhattacharyya et al. [Commun. Stat., Theory Methods 17, No. 11, 3629–3644 (1988; Zbl 0696.62176)] and M. C. Jones [Biometrika 78, No. 3, 511–519 (1991; Zbl 1192.62107)], respectively, for estimating the underlying density \(f\) at a point under a general selection biased model. The asymptotic optimality of \(\widetilde{f}_n\) and \(\widehat{f}_n\) is measured by the corresponding asymptotic minimax mean squared errors under a compactly supported Lipschitz continuous family of the underlying densities. It is shown that, in general, \(\widehat{f}_n\) is a superior local estimate to \(\widetilde{f}_n\) in the sense that the asymptotic minimax risk of \(\widehat{f}_n\) is lower than that of \(\widetilde{f}_n\). The minimax kernels and bandwidths of \(\widehat{f}_n\) are computed explicitly and shown to have simple forms and to depend on the weight functions of the model.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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References:

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