Estimating the Scale Parameter of the Exponential Distribution with Unknown Location

Abstract

Let $X_1,\cdots, X_n$ be i.i.d. random variables each with a density which is $a^{-1} \exp (b - x)$ or 0 according as $x \geqq b$ or $x < b$ where $-\infty < b < \infty, a > 0$ are unknown constants. Let $\bar{X} = n^{-1} \sum X_i$ and $M = \min X_i$. The maximum likelihood estimator of $a$ is $\bar{X} - M$ and is, if quadratic loss is assumed, the best affine equivariant estimator of $a$. It is shown that if loss is measured by any member of a large class of "bowl-shaped" functions which includes quadratic loss, the best affine equivariant estimator is inadmissible. The proof entails an examination of the conditional expected loss given the maximal invariant under the scale group. It is carried out by exhibiting a superior alternative. In the case of quadratic loss, for example, the result is as follows. Given any estimator $u = (\bar{X} - M)T\lbrack M(\bar{X} - M)^{-1}\rbrack$, let $T^\ast(y) = T(y)$ if $y < 0$ and $T^\ast(y) = \min \{T(y), n(n + 1)^{-1}(1 + y)\}$ if $y > 0$. If $T^\ast \neq T$ with positive probability, then the estimator, obtained from $u$ by replacing $T$ by $T^\ast$, has uniformly smaller risk than $u$. Using a generalization of the author's conditions for admissibility [Ann. Math. Statist. 41 (1970) 446-457] a class, $B$, of generalized Bayes estimators within $D$, the class of scale equivariant estimators, are obtained with each member of $B$ admissible in $D$. The improper measures determining members of $B$ have densities on the orbit space $R$, created in the parameter space by the action of the group of scale changes. These prior densities, $g$, satisfy $\int^\infty_1 (t^2g(t))^{-1} dt = \int^{-1}_{-\infty} (t^2g(t))^{-1} dt = \infty$.