Conformal geometry of statistical manifold with application to sequential estimation

TL;DR

This paper introduces a geometric approach to analyze sequential estimation procedures using conformal geometry, providing conditions for covariance minimization and applying the theory to specific statistical models.

Contribution

It develops a dual conformal curvature framework for sequential estimation, extending the theory to multidimensional curved exponential families and validating with numerical examples.

Findings

Conditions for covariance minimization are clarified.

The dual conformal curvature concept is introduced.

Numerical analysis confirms theoretical results.

Abstract

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.