On estimating a bounded normal mean with applications to predictive density estimation

Abstract

For a normally distributed $X\sim N(\mu,\sigma^{2})$ and for estimating $\mu$ when restricted to an interval $[-m,m]$ under general loss $F(|d-\mu|)$ with strictly increasing and absolutely continuous $F$, we establish the inadmissibility of the restricted maximum likelihood estimator $\delta_{\hbox{mle}}$ for a large class of $F$’s and provide explicit improvements. In particular, we give conditions on $F$ and $m$ for which the Bayes estimator $\delta_{BU}$ with respect to the boundary uniform prior $\pi(-m)=\pi(m)=1/2$ dominates $\delta_{\hbox{mle}}$. Specific examples include $L^{s}$ loss with $s>1$, as well as reflected normal loss. Connections and implications for predictive density estimation are outlined, and numerical evaluations illustrate the results.