Geodesic
The structure of the UMVUEs from categorical data
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 3, pp. 597-604

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Let an observation $X$ take finitely many values with probabilities $p_1(\theta),\ldots,p_N(\theta)$ depending on an abstract parameter $\theta\in\Theta$. It is proved that a statistic is a uniformly minimum variance unbiased estimator (UMVUE) if and only if it is measurable with respect to a subalgebra of the finite algebra generated by $X$. In general, this subalgebra is smaller than the minimal sufficient subalgebra for $\theta$ and is explicitly described. It is related to a special partition of a finite set of elements of an abstract linear space.
Keywords: linear space, sufficiency.
Mots-clés : estimation, partition, subalgebra 
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A. Kagan; M. Konikov. The structure of the UMVUEs from categorical data. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 3, pp. 597-604. http://geodesic.mathdoc.fr/item/TVP_2005_50_3_a14/