## Minimaxity of the empirical distribution function in invariant estimation.(English)Zbl 0739.62011

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.

### MSC:

 62C20 Minimax procedures in statistical decision theory 62G07 Density estimation
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