بهترين براوردگرهاي ناريب در صورت عدم وجود بسندگي و كامل بودن.

The best unbiased estimators in the absence of sufficiency and completeness.

Introduction One of the goals of mathematical statistics is to estimate and find a good estimate for an unknown parameter of population. In estimation problem, finding a good estimator is relative, and a criterion must be chosen under which an estimator is good. One of these methods is to find the best estimator under a certain class. If we are looking for the best estimator (the minimum risk estimator under given loss) in the class of unbiased estimators, the estimator is called the minimum risk unbiased estimator, and in the special case under the squared error loss, the risk is converted to variance and the uniformly minimum variance unbiased estimator (UMVUE) is found as an optimal estimator. In some cases, the complete sufficient statistic (CSS) does not exist, there may also be non-fixed parameter functions, which are UMVU estimates. Material and Methods In this study, a simple generalization of the Lehmann-Scheffe theorem (Lehmann and Scheffe, 1950) is proposed in cases where UMVUEs exist but a CSS 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. Conclusion If a CSS exists, every estimable parametric function are UMVUE. Conversely, if each estimable parametric function is an UMVUE, then there will be a CSS under certain conditions. In some cases, although there is not CSS, there may also be non-fixed parametric functions, which are UMVUE. In this paper, generalizations and simple examples are given that UMVUE exists, but not CSS are available. In case there is no CSS and a suitable statistic that is independent of the ancillary statistic is not easily found, conditioning the sufficient statistic on the ancillary statistic will recover the lost information. A UMVUE for an unknown parameter can also be found using a commutative and associative binary operation. It should be noted that completeness and unbiasedness are not characteristic of a statistic or its parametric form, but are characteristic of the family of distributions of a statistic, and deleting even one point of the parameter space can lose the characteristic of completeness. [ABSTRACT FROM AUTHOR]

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