Exponential dispersion models: Second-order minimax estimation of the mean for unknown dispersion parameter

Exponential dispersion models (EDMs) with unknown mean and dispersion parameters are studied with respect to second-order minimax estimation of the mean, for a nonrestricted space of mean values. A modified second-order minimax estimator is presented and some of its asymptotic properties are established. In particular, we provide some necessary and sufficient conditions for such modified estimators to improve on the sample mean. The present results extend those of Landsman [Second order minimax estimation of the mean value for exponential dispersion models. J. Statist. Plann. Inference, 98, 57-71, 2001.] which were given for EDMs with a known dispersion parameter. Applications to the Tweedie class with the power parameter γ,2⩽γ⩽∞, are presented.