This article is cited in 13 papers

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

V. M. Maksimov

Moscow

Abstract: 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).

Received: 14.01.1966

 English version: Theory of Probability and its Applications, 1967, 12:2, 267–280

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