Minimax Prediction of the Empirical Distribution Function

Abstract

Let X 1,…, X n be i.i.d. random variables from an unknown cumulative distribution function F defined on the real line ℝ. No assumptions are made on the unknown F. The problem is to predict the empirical distribution function of a future sample Y = (Y 1,…,Y m ) from the distribution function F on the basis of the observations X 1,…, X n . To solve the problem, a decision theoretic approach is used. In certain situations, when for example a weighted quadratic loss function is used in the prediction problem, the problem of finding the minimax predictor of the empirical distribution function can be reduced to the problem of minimax estimation of the cumulative distribution function F under a composite weighted quadratic loss function. In such a case, a general method of finding minimax predictors of is presented. The method is also useful when no equalizer rule is available. The problem of prediction considered refers also to prediction of the empirical lifetime distribution function and may have some practical significance. For example, the results enable to predict, under the minimax criterion, the number of devices (from among all the m ones) which will fail up to any fixed time t 0.