Minimaxity of the empirical distribution function in invariant estimation (Q1175391)

Consider the problem of invariant (under monotone transformations) estimation of continuous distribution functions with Cramér-von Mises loss function weighted by \(h(t)=t^{-1}(1-t)^{-1},\;t\in(0,1)\). The minimaxity of the empirical distribution function which, as is well- known, is the best invariant estimator in this problem, is proved for sample sizes \(n>2\). A detailed proof is given for the case \(n=3\) and under some additional conditions on the class of estimators. The proof for the general case is only outlined.