Pearson information-based lower bound on Fisher information

TL;DR

This paper introduces a new lower bound on the Fisher information matrix called the Pearson information matrix, which is useful when the data distribution is unknown or intractable, and relates it to estimation accuracy.

Contribution

It derives the Pearson information matrix as a lower bound on Fisher information using moment constraints and links it to the asymptotic covariance of the generalized method of moments.

Findings

The inverse PIM matches the asymptotic covariance of GMM estimators.

The PIM provides a practical lower bound when the data distribution is unknown.

The bound relates to properties of misspecified data distributions.

Abstract

The Fisher information matrix (FIM) plays an important role in the analysis of parameter inference and system design problems. In a number of cases, however, the statistical data distribution and its associated information matrix are either unknown or intractable. For this reason, it is of interest to develop useful lower bounds on the FIM. In this lecture note, we derive such a bound based on moment constraints. We call this bound the Pearson information matrix (PIM) and relate it to properties of a misspecified data distribution. Finally, we show that the inverse PIM coincides with the asymptotic covariance matrix of the optimally weighted generalized method of moments.