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April, 1972 Transformation Groups and Sufficient Statistics
J. Pfanzagl
Ann. Math. Statist. 43(2): 553-568 (April, 1972). DOI: 10.1214/aoms/1177692636


Let $(X, \mathscr{U})$ be a pathwise connected and locally connected topological space with countable base, and $(\Theta, \mathscr{W})$ be a connected and locally connected continuous transformation group on $X$ which is Abelian. Let $\mathscr{A}$ be the smallest $\sigma$-field containing all open sets and $P \mid \mathscr{A}$ a probability measure such that $P(U) > 0$ for every open set $U \neq \varnothing$. For every $\vartheta \in \Theta$ let $P_\vartheta$ denote the probability measure generated by the transformation $\vartheta$, i.e. $P_\vartheta(A): = P(\vartheta^{-1} A), A \in \mathscr{A}$. Assume that $P_\vartheta$ admits a continuous density relative to $P$ for every $\vartheta \in \Theta$. Assume finally that for some sample size $n > 1$ there exists a real-valued, continuous statistic $T_n$ which is equivariant (i.e. $T_n(x_1, \cdots, x_n) = T_n(y_1, \cdots, y_n)$ implies $T_n(\vartheta x_1, \cdots, \vartheta x_n) = T_n(\vartheta y_1, \cdots, \vartheta y_n)$ for all $\vartheta \in \Theta$) and sufficient for $P_{\vartheta^n,} \vartheta \in \Theta$. Under these assumptions there exists a real-valued, continuous statistic $S$ on $X$ which is sufficient for $P_\vartheta, \vartheta \in \Theta$, such that the distribution of $S$ is either the location parameter family of normal distributions with variance 1 or a scale parameter family of gamma distributions. In a nutshell: Among the families of distributions which are generated by Abelian transformation groups, and which fulfill certain regularity conditions, the location parameter family of normal distributions and the scale parameter families of gamma distributions are essentially the only ones admitting for some sample size greater than one a sufficient statistic which is real valued, continuous and equivariant.


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J. Pfanzagl. "Transformation Groups and Sufficient Statistics." Ann. Math. Statist. 43 (2) 553 - 568, April, 1972.


Published: April, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0237.62004
MathSciNet: MR300359
Digital Object Identifier: 10.1214/aoms/1177692636

Rights: Copyright © 1972 Institute of Mathematical Statistics


Vol.43 • No. 2 • April, 1972
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