Second order minimax estimation of the mean

In this study we consider the problem of the improvement of the sample mean in the second order minimax estimation sense for a mean belonging to an unrestricted mean parameter space R+. We solve this problem for the class of natural exponential families (NEF's) whose variance functions (VF's) are regular at zero and at infinity. Such a class of VF's (or NEF's) is huge and contains (among others): Polynomial VF's (e.g., quadratic VF's in the Morris class, cubic VF's in the Letac&Mora class and VF's in the Hinde-Demétrio class); VF's belonging to the Tweedie class with power VF's, VF's belonging to the Babel class and many others. Moreover, we show that if the canonical parameter space of the corresponding NEF is R (which is obviously the case if the support of the NEF is bounded), then the sample mean as an estimator of the mean cannot be further improved. This work presents an original constructive methodology and provides with constructive tools enabling to obtain explicit forms of the second order minimax estimators as well as the forms of the related weight functions. Our work establishes a substantial generalization of the results obtained so far in the literature. Illustrations of the resulting methods are provided and a simulation-based analysis is presented for the negative binomial case.