Necessary and Sufficient Statistics for the Family of Shifts of Probability Distributions on Continious Bicompact Groups

V. M. Maksimov

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 2, pp. 307-321

Résumé

It is shown that if the family of probability distributions $p_y(x)=p(xy)$ on a bicompact group $G$, where $x\in G$ and $y\in G$, has nontrivial sufficient statistics for the parameter $у$ then the density $p(x)$ may be written in the form $p(x)=\exp\Bigl(\lambda+\sum_{k=1}^sc_k\varphi_k(x)\Bigr)$ where $1,\varphi_1(x),\dots,\varphi_s(x)$ is a basis of the set of entries of the matrix $\{g_{ij}(x)\}$ of a certain real finite-dimensional representation of group $G$. The case when $p(x)$ may be equal to zero is also considered (here we deal mainly with the case when $G$ is the circle group).