Quadratic estimators of quadratic functions of normal parameters
When, for \(n\geq 2\), \(X_ 1,...,X_ n\) is a normal random sample with unknown mean \(\xi\) and unknown standard deviation \(\sigma\), the estimation problem of the form \(\theta =\xi^ 2+b\sigma^ 2\) under quadratic loss is considered, where b is a given non-negative number. The author discusses the admissibility of the estimator \[ \delta_ 0(\bar X,S)=\bar X^ 2+S^ 2(bn-1)\{n(n+1)\}^{-1} \] for the scaled quadratic loss \(L(\xi,\sigma,\delta)=(\theta -\delta)^ 2\sigma^{-4}\), under which \(\delta_ 0\) has a constant risk, where \(\bar X=\sum^{n}_{i=1}X_ i/n\) and \(S^ 2=\sum^{n}_{i=1}(X_ i-\bar X)^ 2\). In fact the estimator \(\delta_ 0\) is shown to be inadmissible in the case when \(n^{-1}\leq b<1+2n^{-1}\). It is also shown that \(\delta_ 0\) is admissible if and only if \(1+2n^{-1}\leq b<2+3n^{- 1}\), and that \(\delta_ 0\) is a generalized Bayes estimator with respect to a prior which admits a good approximation, in terms of posterior risk, by probability priors.
scientific article; zbMATH DE number 3986392
scientific article