MINIMAX INVARIANT ESTIMATOR OF A CONTINUOUS DISTRIBUTION FUNCTION

QIQING YU

Department of Applied Mathematics, State University of New York, Stony Brook, NY 11794, U.S.A.

(Received February 1, 1991; revised August 14, 1991)

Abstract. Consider the problems of the continuous invariant estimation of a distribution function with a wide class of loss functions. It has been conjectured for long that the best invariant estimator is minimax for all sample sizes n ≥ 1. This conjecture is proved in this short note.

Key words and phrases: Minimaxity, invariant estimator, nonparametric estimator, product measure, Lebesgue measure, uniform distribution on a set.