On the minimax property for R-estimators of location
The author considers the problem of estimating the unknown location parameter \(\theta\) based on a random sample from F(x-\(\theta)\), where F is unknown member of \({\mathcal F}_{\epsilon}=\{F|\) F is symmetric about 0 and \(\sup_{x}| F(x)-\Phi(x)| \leq \epsilon\), \(0<\epsilon<1\) fixe\(d\}\), where \(\Phi\) (x) denotes the standard normal distribution function. While for M- and L-estimators of location the answer for the question ''Does the minimax property hold?'' is known, this was not the case for R-estimators and the present author filled this gap. More precisely, following theorem is proved. Theorem. Let \(\epsilon\), \(0<\epsilon<1\), be fixed and \({\mathcal F}_{\epsilon}\) defined as above. Let \(F_ 0\) be the distribution minimizing the Fisher information I(F) over \({\mathcal F}_{\epsilon}\) and let \(J_ 0\) be the asymptotically efficient score function corresponding to \(F_ 0\). Then the minimax property holds, i.e. \(\sup \{V(J_ 0,F):\) \(F\in {\mathcal F}_{\epsilon}\}=V(J_ 0,F_ 0)\), where \[ V(J,F)=\int J^ 2(F(x))f(x)dx/\{\int J'(F(x)).f^ 2(x)dx\}^ 2. \]
scientific article; zbMATH DE number 3862230