Inadmissibility of the Best Invariant Estimator of Extreme Quantiles of the Normal Law Under Squared Error Loss

Abstract

Suppose that independent normally distributed random vectors, $X^{n\times1}$ and $Y^{k\times1}$, are observed with $E(X) = 0, E(Y) = \mu, \operatorname{Cov}(X) = \sigma^2I$, and $\operatorname{Cov}(Y) = \sigma^2I$. It is known [2, 5] that the best invariant estimator of $\mu$ is admissible if $k \leqq 2$ and inadmissible if $k > 2$. It is also known [1, 7] that the best invariant estimator of $\sigma$ is inadmissible. In this paper, these results are extended to show that the best invariant estimator of $\theta = A\mu + \eta\sigma$, for a given matrix $A$ and a given vector $\eta$, is inadmissible if $|\eta|$ is sufficiently large (when $k = 1, A = 1, \theta$ is a quantile).