A differential geometric approach to statistical inference on the basis of contrast functionals (Q579783)

In this paper, we consider contrast functionals on the space of all probability measures equivalent to each other. Many examples of the contrast functional have been proposed and estimation methods based on them, called the minimum contrast estimation methods, have been investigated since the theory of estimation was initiated by R. A. Fisher. It is shown that a contrast functional generates a conjugate metric structure with a Riemannian metric and a conjugate pair of affine connections on the space. We show that this structure explains some properties of the minimum contrast estimator. In particular, the explicit formulas for the limiting information loss for the estimators are given in covariance structure models. Moreover we propose a generalized scoring method for seeking the minimum contrast estimates. It is shown that the convergence of the algorithm is affected by two geometric quantities which can be expressed in the conjugate metric structure.