Conformal Geometry of Sequential Test in Multidimensional Curved Exponential Family

TL;DR

This paper introduces a differential geometric approach to analyze sequential testing procedures in multidimensional curved exponential families, focusing on conformal geometry and curvature-related variables.

Contribution

It develops a new geometric framework for understanding sequential tests in curved exponential families, extending previous conformal geometry results to multidimensional cases.

Findings

Curvature-type variables characterize when a manifold is an exponential family.

The method is applied to von Mises-Fisher and hyperboloid models.

Theoretical insights are supported by numerical experiments.

Abstract

This article presents a differential geometrical method for analyzing sequential test procedures. It is based on the primal result on the conformal geometry of statistical manifold developed in Kumon, Takemura and Takeuchi (2011). By introducing curvature-type random variables, the condition is first clarified for a statistical manifold to be an exponential family under an appropriate sequential test procedure. This result is further elaborated for investigating the efficient sequential test in a multidimensional curved exponential family. The theoretical results are numerically examined by using von Mises-Fisher and hyperboloid models.