Minimaxity of the empirical distribution function in invariant estimation
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.
finite sample size invariant decision problem
minimaxity of the empirical distribution
scientific article; zbMATH DE number 11543