Locally minimax efficiency of nonparametric estimates of square- integrable densities (Q1324890)

The paper gives a necessary and sufficient condition for the existence of the estimate \(f^*\) of the probability density \(f\), such that for any function \(g\in W^ 0\) \[ \sup_{f\in W_ g} E_ f\| f^*-f\|^ 2_ 2 \sim\inf_{\widehat {f}} \sup_{f\in W_ g} E_ f\| \widehat{f}- f\|^ 2_ 2. \] Here \(W^ 0\) is the interior of \(W\subseteq L_ 2\), a subset of all square-integrable probability densities having compact support. Additionally \(f^*\) is given as a minimizer of the first component of a random functional \(G(p,g,h)\) on \(L_ 2\times W\times H\), where \(H\) is the set of all selfadjoined operators from \(L_ 2\) in \(L_ 2\) and \(g\) and \(h\) depend suitably on \(p\).