On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters. Let X = (X1, X2, ..., Xn) be an observation from a multivariate exponential distribution with that natural parameter [Theta] = ([Theta]1, [Theta]2, ..., [Theta]n). Let [theta]x* be the posterior mode. Sufficient conditions are presented for the distribution of [Theta] - [theta]x* given X = x to converge to a multivariate normal with mean vector 0 as x tends to infinity. These same conditions imply that E([Theta] X = x) - [theta]x* converges to the zero vector as x tends to infinity. The posterior for an observation X = (X1, X2, ..., Xn is considered for a location vector [Theta] = ([Theta]1, [Theta]2, ..., [Theta]n) as x gets large along a path, [gamma], in Rn. Sufficient conditions are given for the distribution of [gamma](t) - [Theta] given X = [gamma](t) to converge in law as t --> [infinity]. Slightly stronger conditions ensure that [gamma](t) - E([Theta] X = [gamma](t)) converges to the mean of the limiting distribution. These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let [delta] be a Bayes estimator for a loss function of this type. Generally, if the distribution of [Theta] - E([Theta] X = [gamma](t)) given X = [gamma](t) converges in law to a symmetric distribution as t --> [infinity], it is shown that [delta]([gamma](t)) - E([Theta] X = [gamma](t)) --> 0 as t --> [infinity].