Conditional natural exponential families

Abstract

Let F={P[theta];[theta][set membership, variant][Theta]} be a natural exponential family on and let H be an exposed face of the closed convex hull of the F support. The aim of this paper is to study the asymptotic behavior of the law P[theta]+[lambda]u as [lambda] increases to +[infinity], for all u exterior normal vector of H. We obtain a degenerate weak limit P[theta],+[infinity] concentrated on the face H. The relation between the conditional law of P[theta] given H and P[theta],+[infinity] is established and some illustrating examples are given.Conditional law Convex Exponential family Face Means domain Variance function