Asymptotic expansions of mean absolute error of uniformly minimum variance unbiased and maximum likelihood estimators on the one-parameter exponential family model of lattice distributions

The paper considers a model of duplicate sampling with the fixed size $n$ from a lattice distribution belonging to the natural one-parameter exponential family. Asymptotic expansions of the mean absolute errors of the uniformly minimum variance unbiased estimator (UMVUE) and the maximum likelihood estimator (MLE) of the given parametric function are obtained in the case of infinite size of the sample. The case when $G'[a]=0$ and $G''[a] \neq 0 $ was studied separately. The relative error in calculating the difference in the mean absolute error UMVUE and MLE was evaluated in the case of the Poisson distribution for the two parametric functions. This error was received via the asymptotic expansions. It was found that the asymptotic results with a sufficiently large sample size allows one to compare UMVUE and MLE using such indicator of quality assessment as the mean absolute error.