A NOTE ON THE BEST INVARIANT ESTIMATOR OF A DISTRIBUTION FUNCTION UNDER THE KOLMOGOROV-SMIRNOV LOSS

ZHIQIANG CHEN AND ESWAR PHADIA

Department of Mathematics, William Paterson College of New Jersey, Wayne, NJ 07470, U.S.A.

(Received February 23, 1996; revised May 13, 1996)

Abstract. For the invariant decision problem of estimating a continuous distribution function with the Kolmogorov-Smirnov loss within the class of ‘proper’ distribution functions, it is proved that the sample distribution function is the best invariant estimator only for the sample size n = 1 and 2. Further it is shown that the best invariant estimator is minimax. Exact jumps of the best invariant estimator are derived for n ≤ 4.

Key words and phrases: Best invariant estimator, Kolmogorov-Smirnov loss, minimaxity.