Limit distributions of unbiased estimators in natural exponential families

Abstract

We obtain the possible limit distributions of unbiased estimators of functions of the parameter of a natural exponential family. The limit distribution depends on <$>j<$>, the order of the first non-zero derivative at the true (but usually unknown) value of the parameter. We show that if <$>j \\geq 2<$> then the umvu and the maximum likelihood estimators are not asymptotically equivalent.