Optimal estimation of scale parameters (Q1087264)

Let observations be represented by the vector \(X=(X_ 1,...,X_ n)\) of nonnegative random variables which may be arbitrarily correlated and possibly nonstationary. Let the densities of \(X_ i's\) be of the form \(p_{\theta}(x_ i)=\theta f_ i(\theta x_ i)\), if \(x_ i\geq 0\), and \(p_{\theta}(x_ i)=0\), if \(x_ i<0\), \(\theta >0\), \(i=1,...,n\). Assume that the joint probability function of X is of the form \(p_{\theta}(x^ n)=\theta^ nf^ n(\theta x^ n)\), if \(\min (x_ 1,...,x_ n)\geq 0\), and \(p_{\theta}(x^ n)=0\), if \(\min (x_ 1,...,x_ n)<0\), \(x^ n\in {\mathbb{R}}^ n_+.\) Let \({\mathfrak U}\) be the class of estimators \({\hat \theta}_ n\) such that \({\hat \theta}_ n(cx^ n)=c^{-1}{\hat \theta}_ n(x^ n)\) for every \(c>0\) and for every \(x^ n\in {\mathbb{R}}^ n_+\) such that \(\min (x_ 1,...,x_ n)\geq 0\). The author looks for the estimator \({\hat \theta}_ n(x^ n)\) of \(\theta\) which is proportional to the inverse of the observation mean and which minimizes the conditional risk uniformly in \(\theta >0\) over \({\mathfrak U}\). More precisely, he works with \[ {\hat \theta}_{n_ 0}(x^ n)= \min_{{\hat \theta}_ n\in {\mathfrak U}}E_{\theta}| {\hat \theta}_ n-\theta |^{\alpha},\quad \alpha >0. \] It is shown that \({\hat \theta}_{n_ 0}(x^ n)\) can be obtained by the generalized Bayesian approach. The general results are applied to a random sample from a gamma distribution to construct point and interval estimators for the scale parameter in order to show the efficiency of this approach.