OPTIMUM BANDWIDTHS AND KERNELS FOR ESTIMATING CERTAIN DISCONTINUOUS DENSITIES

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 f̂_{n} to rectify certain drawbacks of f_{n}. This paper gives the asymptotically optimum bandwidth and kernel for f̂_{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 f̂_{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.