Transformation Groups and Sufficient Statistics

Abstract

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.