Inadmissibility Results for the Best Invariant Estimator of Two Coordinates of a Location Vector

Abstract

Let $X = (X_1, X_2, X_3)$ be a random vector with density $f(x - \theta)$, where $\theta = (\theta_1, \theta_2, \theta_3)$ is unknown. It is desired to estimate $(\theta_1, \theta_2)$ using an estimator $(\delta_1(X), \delta_2(X))$, and under a loss function $L(\delta_1 - \theta_1, \delta_2 - \theta_2)$. (Note that $\theta_3$ is a nuisance parameter.) Under certain conditions on $f$ and $L$, it is shown that the best invariant estimator of $(\theta_1, \theta_2)$ is inadmissible.