Regularity conditions for an improper prior function to be regarded as a virtually proper prior density are proposed, and their implications are discussed. The two regularity conditions require that a prior function is defined as a limit of a sequence of proper prior densities and also that the...

An empirical Bayes method for a gamma regression model with a slope parameter [beta] common through multiple strata is proposed and examined. Assuming that the number of strata is fairly large, we attempt to modify a usual empirical Bayes method so as to yield an improved estimator of [beta]....

The conjugate prior for the exponential family, referred to also as the natural conjugate prior, is represented in terms of the Kullback-Leibler separator. This representation permits us to extend the conjugate prior to that for a general family of sampling distributions. Further, by replacing...

Consider a class of estimators of a mean vector, indexed by a parameter. We introduce a pair of estimating equations for the parameter and the variance in the normal distribution. The equations provide us with an interpretation of the empirical Bayes method for smoothing and the James-Stein...