On joint completeness: sampling and Bayesian versions, and their connections
Ernesto San Martı́n, Michel Mouchart · 2007
Cramer, Kamps and Schenk (Statist. Decisions, 2002) established conditions under which a family of joint distributions of two independent statistics is complete, and related their result with a previous one of Landers and Rogge (Scand. J. Statist., 1976). We first propose, within a sampling theory framework, a modification of Cramer, Kamps and Schenk's (2002) generalization, paying a particular attention to the concept of completeness with respect to a function of a parameter. Next, after reviewing Bayesian completeness on the sample space, it is shown that Landers and Rogge's (1976) theorem can be extended to a Bayesian framework. A Bayesian version of Cramer, Kamps and Schenk's (2002) theorem is also provided. These results are illustrated with examples in both a normal and a discrete experiment. Finally, taking advantage of the formal symmetry between parameters and observations in a Bayesian experiment, we show that Landers and Rogge type theorems are useful when analysing Bayesian identifiability of structural models often used for modelling individual data.