Conformal geometry of statistical manifold with application to sequential estimation

We present a geometrical method for analyzing sequential estimating procedures. It is based on the design principle of the second-order efficient sequential estimation provided in Okamoto, Amari and Takeuchi (1991). By introducing a dual conformal curvature quantity, we clarify the conditions for the covariance minimization of sequential estimators. These conditions are further elabolated for the multidimensional curved exponential family. The theoretical results are then numerically examined by using typical statistical models, von Mises-Fisher and hyperboloid models.