On Bounded Completeness and the $L_1$-Denseness of Likelihood Ratios

TL;DR

This paper provides a direct proof of a classical result linking bounded completeness to $L^1$-denseness of likelihood ratios, with implications for finite and infinite-dimensional statistical experiments.

Contribution

It offers a new, straightforward proof of a known characterization of bounded completeness and explores its application to finite and infinite-dimensional experiments.

Findings

Bounded completeness is characterized by $L^1$-denseness of likelihood ratios.

Infinite-dimensional experiments are boundedly complete iff finite-dimensional subexperiments are.

The proof simplifies understanding of bounded completeness in statistical models.

Abstract

The classical concept of bounded completeness and its relation to sufficiency and ancillarity play a fundamental role in unbiased estimation, unbiased testing, and the validity of inference in the presence of nuisance parameters. In this short note, we provide a direct proof of a little-known result by \cite{Far62} on a characterization of bounded completeness based on an $L^1$ denseness property of the linear span of likelihood ratios. As an application, we show that an experiment with infinite-dimensional observation space is boundedly complete iff suitably chosen restricted subexperiments with finite-dimensional observation spaces are.