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MINIMAX KERNELS FOR DENSITY ESTIMATION

WITH BIASED DATA

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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 ^{~}*f*_{n} and ^{^}*f*_{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
^{^}*f*_{n} and ^{~}*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, ^{^}*f*_{n} is a superior local estimate than
^{~}*f*_{n} in the sense that the asymptotic minimax risk of
^{^}*f*_{n} is lower than that of ^{~}*f*_{n}. The minimax kernels
and bandwidths of ^{^}*f*_{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.

**Source**
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