The best unbiased estimators in the absence of sufficiency and completeness

Authors
Mehdi Shams 1
Gholamreza Hesamian 2
1 University of Kashan
2 Payame Noor University
Abstract
In this study, a simple generalization of the Lehmann-Scheffe theorem is proposed in cases where UMVUEs exist but a sufficiently complete statistics does not exist. Also, another method is introduced based on the group action. In this method, UMVUE for the unknown parameter is found using a commutative and associative binary operation. Finally, the motivation for using the words "completeness" and "unbiasedness" is expressed in such a way that completeness and unbiasedness are not characteristic of a statistic or its parametric form, but a family of distributions of a statistic, and deleting even one point of the parameter space may change the completeness.
Keywords
Complete sufficient statistics
minimal sufficient statistics
uniformly minimum variance unbiased estimator
Fisher information
binary operation
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