A Cram\'er-Rao inequality for non differentiable models

TL;DR

This paper introduces a new variance lower bound for unbiased estimators applicable to non-differentiable models, extending the classical Cramér-Rao bound and providing sharper efficiency bounds.

Contribution

It develops a Cramér-Rao type inequality that does not require model differentiability, improving bounds in non-smooth statistical models.

Findings

The new bound is always greater than the classical Cramér-Rao bound in smooth models.

The bound applies to non-differentiable models, broadening the scope of efficiency analysis.

It provides a sharper variance lower bound for unbiased estimators.

Abstract

We compute a variance lower bound for unbiased estimators in specified statistical models. The construction of the bound is related to the original Cram\'er-Rao bound, although it does not require the differentiability of the model. Moreover, we show our efficiency bound to be always greater than the Cram\'er-Rao bound in smooth models, thus providing a sharper result.