Inadmissibility Results for Generalized Bayes Estimators of Coordinates of a Location Vector

Abstract

Let $X = (X_1, \cdots, X_n)$ be an $n$-dimensional random vector with density $f(x - \theta)$. Assume the loss incurred in estimating $\theta$ by $d$ is $L(d - \theta)$ and is convex. Let $F$ be a generalized prior density. The question of inadmissibility of generalized Bayes estimators of $\theta$ is considered. As in Brown (1974c), the crucial role played by the "moment structure" in determining inadmissibility results is indicated (moments defined as the quantities $m_{i,j(1), \cdots, j(l)} = \int \prod^l_{k=1} x_{j(k)} \cdot \lbrack(\partial/\partial x_i)L(x)\rbrack f(x) dx)$. Detailed inadmissibility results are given for a particular moment structure, one which arises most naturally in trying to estimate a single coordinate or a linear combination of coordinates of $\theta$. For example, suppose $\theta_1$ is to be estimated by the generalized Bayes estimator $\delta_F$. (Thus $\theta_2, \cdots, \theta_n$ are nuisance parameters.) Under certain conditions, the most important being that the moment structure be of a certain form, it is shown that if there exist constants $\lambda > 0$ and $T > 0$ such that $\delta_F(x) \geqq x_1 - (n - 3 - \lambda)\big/ \big\lbrack 2x_1 \int\big(\frac{\partial^2}{\partial x_1^2} L(x)\big) f(x) dx\big\rbrack \quad \text{for} x_1 > T,$ then $\delta_F$ is inadmissible. Thus, for example, under certain quite general assumptions, the best invariant estimator, $\delta_0(x) = X_1$, of the first coordinate of a location vector is inadmissible if $n \geqq 4$.