A note on insufficiency and the preservation of Fisher information

TL;DR

This paper explores why, in certain mixture models, insufficient statistics can still preserve Fisher information, using differentiability in quadratic mean and linking to Le Cam's theory.

Contribution

It offers a new explanation for Fisher information preservation in mixture models through regularity properties and connects this phenomenon to Le Cam's convergence theory.

Findings

Insufficient statistics can preserve Fisher information in specific mixture models.

Differentiability in quadratic mean explains Fisher information preservation.

Connections to Le Cam's theory provide deeper insight into experiment convergence.

Abstract

Kagan and Shepp (2005, Amer. Statist.) presented an elegant example of a mixture model for which an insufficient statistic preserves Fisher information. This note uses the regularity property of differentiability in quadratic mean to provide another explanation for the phenomenon they observed. Some connections with Le Cam's theory for convergence of experiments are noted.