MINIMAX KERNELS FOR DENSITY ESTIMATION WITH BIASED DATA

COLIN O. WU AND ANDREW Q. MAO

Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218, U.S.A.

(Received August 5, 1994; revised September 25, 1995)

Abstract. This paper considers the asymptotic properties of two kernel estimates f̃_{n} and f̂_{n}, 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 f̂_{n} and f̃_{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, f̂_{n} is a superior local estimate than f̃_{n} in the sense that the asymptotic minimax risk of f̂_{n} is lower than that of f̃_{n}. The minimax kernels and bandwidths of f̂_{n} 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.