On the Minimax Value in the Scale Model with Truncated Data

Abstract

Let $X$ be a positive random variable with Lebesgue density $f_\theta(x)$, where $\theta$ is the scale parameter, and let $Y$ be a positive random variable independent of $X$. We consider two models of truncation: the LHS model, where the data consist only of those observations of $X$ for which $X > Y$; and the RHS model, where the data consist of those observations of $X$ for which $X \leq Y$. Consider the problem of estimating $\theta^s, s \neq 0$, under a normalized squared error loss function. It is shown that under appropriate assumptions, if $f_1(\cdot)$ varies regularly at 0 (or $+ \infty$), then the minimax value in the RHS (LHS) model is equal to 1 for arbitrarily large sample size. This implies the existence of trivial minimax and admissible estimators, which do not depend on the sample at all, in contrast with the scale model without truncation.