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

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.