Estimating Common Mean in Heteroscedastic Variances Model

Bayes estimators for the unknown mean against a reference, non-informative prior distribution for both the mean and independent variances are derived. I entertain the scenario with two groups of observables with the same unknown mean. The unknown variances of the the first group are not supposed to be equal or to be restricted; the second homeogeneous group of observations all have the same unknown variance. Under the normality condition, these procedures turn out to have a very explicit form of the weighted average with data-dependent weights that admit of a very clear interpretation. The approximate formulas for the variance of the considered estimators and their limiting behavior are also examined. The related “self-dual” orthogonal polynomials and their properties are examined. Recursive formulas for estimators on the basis of these polynomials are developed.