Higher-order asymptotic expansions of unbiased estimators and their variances on the one-parameter exponential family model

The paper considers a model of duplicate sampling with the fixed size $n$ from a distribution belonging to the natural one-parameter exponential family. A limiting behavior of the uniformly minimum variance unbiased estimator (UMVUE) of the given parametric function and the UMVUE variance of this estimator is studied in the case of infinite size of the sample. Higher-order asymptotic expansions are obtained for functions defining unbiased estimators and variances of these estimators. The results are presented for both the canonical parameterization and the mean parameterization.