Minimax and Gamma-minimax estimation for functions of the bounded parameter of a scala parameter family under L_p-loss

Kurzfassung/Abstract

Let $X$ be a random variable with density $f(t,\theta)=\theta g(t\theta)$, $t\in R\sp +$ or $R$ where the scale parameter $\theta\in I=[\theta\sb 0,(1+m)\theta\sb 0]$, $\theta\sb 0>0$, $m>0$, i.e. $\theta$ lies in a bounded interval $I$. The problem is to estimate a strictly monotonic parametric function $h(\theta)$ under the loss function $L\sb p(\theta,d)=\vert h(\theta)-d\vert\sp p$ where $p\geq 2$ and $d$ is an estimator. The author obtains minimax and $\Gamma$-minimax (minimax under Gamma prior) estimators under the $L\sb p$-loss if the interval $I$ is small enough. It is shown that these minimax estimators are Bayes with respect to a two-point prior.