(Received August 5, 1994; revised September 25, 1995)
Abstract. This paper considers the asymptotic properties of two kernel estimates ~fn and ^fn, which have been proposed by Bhattacharyya et al. (1988, Comm. Statist. Theory Methods, A17, 3629-3644) and Jones (1991, Biometrika, 78, 511-519), respectively, for estimating the underlying density f at a point under a general selection biased model. The asymptotic optimality of ^fn and ~fn 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, ^fn is a superior local estimate than ~fn in the sense that the asymptotic minimax risk of ^fn is lower than that of ~fn. The minimax kernels and bandwidths of ^fn are computed explicitly and shown to have simple forms and depend on the weight functions of the model.
Key words and phrases: Kernel density estimate, minimax mean squared error, minimax kernel, bandwidth, weighted distribution, selection biased data.