Minimax invariant estimator of a continuous distribution function (Q1260727)

Let \(X_ 1,X_ 2,\dots,X_ n\) be a random sample from an unknown continuous distribution function \(F\). Let \(A=\{a(t): a(t)\) is a nondecreasing function from \((0,1)\) into \([0,1]\}\) be the action space for the estimation of \(F\). The problem under consideration is the invariant estimation of \(F\) under the loss function \[ L(F,a)=\int| F(t)- a(t)|^ k h(F(t))\,dF(t) \] with \(k\geq 1\) and \(h(t)\geq 0\). It has long been conjectured that the best invariant estimator of \(F\) is minimax for all sample sizes \(n\geq 1\). The author proves this conjecture under the preceding wide class of loss functions.