Minimum Risk Invariant Estimators of a Continuous Cumulative Distribution Function

The problem of nonparametric minimum risk invariant estimation has engaged a good deal of attention in the literature and minimum risk invariant estimators (MRIE's) have been constructed for some special statistical models. We present a new and simple method of obtaining the MRIE's of a continuous cumulative distribution function (cdf) under a general invariant loss function. All the MRIE's, which are known from the literature, can be constructed by the method presented in the article, in particular, under the weighted quadratic, LINEX and entropy loss functions. This method enables also to construct the MRIE's in nonparametric statistical models which have not been considered until now. In particular, considering a family of nonparametric precautionary loss functions, a new class of MRIE's of the cdf has been found. We also give some general remarks on obtaining the MRIE's and a review concerning minimaxity and admissibility of MRIE's.