Sankhya: The Indian Journal of Statistics

2007, Volume 69, Pt. 4, 780--807

On Joint Completeness: Sampling and Bayesian Versions, and Their Connections

Ernesto San Mart\'{\i}n, Pontificia Universidad Cat\'olica de Chile, Santiago, Chile
Michel Mouchart, Universit\'e catholique de Louvain, Louvain-la-Neuve, Belgium

SUMMARY. Cramer, Kamps and Schenk ({\it 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 ({\it 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 {\it 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.

AMS (2000) subject classification. Primary 62H30, 62-07.

Key words and phrases. Completeness, variation-free parametrization, Rasch Poisson counts model, identification.

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