On Bounded Completeness and the L_1-Denseness of Likelihood Ratios

08/02/2023

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by   Marc Hallin, et al.

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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 <cit.> 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.