(Received August 22, 1990; revised June 21, 1991)
Abstract. Rosenblatt and Parzen proposed a well-known estimator fn for an unknown density function f, and later Schuster suggested a modification ^fn to rectify certain drawbacks of fn. This paper gives the asymptotically optimum bandwidth and kernel for ^fn 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 fn and ^fn 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.
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