Inference for the exponential family having a high-dimensional parameter

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The inferential procedure of a high-dimensional parameter is one of the most appealing research subjects in the theoretical and the practical points of view. The present research focuses on the empirical Bayes method in terms of a natural conjugate prior and the application of the theory of estimating function. A special emphasis is placed on the Pythagorean relationship observed in the parameter space of the exponential family.A significant result of the present research project is preformed by extending a natural conjugate prior. We first note that a natural conjugate prior is expressed in terms of the Kullback-Leibler separator. Thus a straightforward extension is possible by replacing the separator by its dual form. Such a prior is called a mean conjugate prior. An attractive property of this prior is that the sampling density and a prior density are common in many familiar cases. Other possible families of prior distribution are also investigated. Another result concerns the relation of an electrostatic quantity, specifically Coulomb potential, with the Pythagorean relationship. More precisely, the maximum likelihood estimator of a multi-dimensional location parameter is improved by shifting it at the step size given by the gradient of the logarithmic Coulomb potential. In light of familiarity of the potential, this result may suggest that further strong relations between the physical and the information sciences can be pursued in the future.At the latest stage the derivation of the link function from the assumption of the strong unbiasedness of the score function was successfully conducted. The derivation shows a special role of the logarithmic link function in the generalized linear model. It is expected that the result will be fully used in studying the generalized linear model with many strata.