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OPTIMUM BANDWIDTHS AND KERNELS FOR ESTIMATING

CERTAIN DISCONTINUOUS DENSITIES

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B. K. GHOSH AND WEI-MIN HUANG

*Department of Mathematics, Lehigh University, Bethlehem, PA 18015, U.S.A.*
(Received August 22, 1990; revised June 21, 1991)

**Abstract.**
Rosenblatt and Parzen proposed a well-known estimator
*f*_{n}
for an unknown density function *f*, and later Schuster suggested a modification
^{^}*f*_{n} to rectify certain drawbacks of *f*_{n}. This paper gives the
asymptotically
optimum bandwidth and kernel for ^{^}*f*_{n} under the standard measure of IMSE
when *f*
is discontinuous at one or both endpoints of its support. We also consider an
alternative definition of the IMSE under which the optimum bandwidths and
kernels
for *f*_{n} and ^{^}*f*_{n} are derived. The latter supplement van Eeden's results.

*Key words and phrases*:
Estimation of discontinuous densities,
alternative notions of IMSE, modified kernel density estimates, optimal
bandwidths,
optimal kernels.

**Source**
( TeX ,
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