On bounded completeness and the L1-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 Farrell (1962) 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 sub experiments with finite dimensional observation spaces are.