Generalized Bayes Solutions in Estimation Problems

Abstract

In estimation problems where the parameter space is not compact the class of Bayes solutions $(\mathscr{B})$ is usually not a complete class and it is necessary to take the closure (in a suitable sense) of $(\mathscr{B})$ to obtain a complete class. When the parameter to be estimated is that of an exponential density the limits of Bayes solutions can be characterized as generalized Bayes solutions in the sense that they minimize a posteriori risk where the a priori distribution may have infinite variation (theorem and corollaries in Section 2). The extent to which exponential densities are necessary for this characterization and some consequences of this characterization are contained in a series of remarks at the end of Section 2. In Section 1 we motivate the ideas by obtaining the above characterization for the problem of estimating the mean of a normal distribution with the parameter space restricted to $\lbrack0, \infty)$ and with squared-error loss function.