Fisher Information of Scale

TL;DR

This paper introduces a new definition of Fisher information of scale for probability distributions on the real line, establishing its properties and connections to classical concepts like differentiability and asymptotic normality.

Contribution

It defines Fisher information of scale as a supremum, proves its mathematical properties, and links it to classical statistical regularity conditions.

Findings

Fisher information of scale is weakly lower semicontinuous and convex.

It is finite under classical density assumptions.

Finite Fisher information of scale implies L_2-differentiability and local asymptotic normality.

Abstract

Motivated by the information bound for the asymptotic variance of M-estimates for scale, we define Fisher information of scale of any distribution function F on the real line as a suitable supremum. In addition, we enforce equivariance by a scale factor. Fisher information of scale is weakly lower semicontinuous and convex. It is finite iff the usual assumptions on densities hold, under which Fisher information of scale is classically defined, and then both classical and our notions agree. Fisher information of scale finite is also equivalent to L_2-differentiability and local asymptotic normality, respectively, of the scale model induced by F.