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 Teor. Veroyatnost. i Primenen., 1966, Volume 11, Issue 3, Pages 483–494 (Mi tvp642)  A systematic theory of exponential families of probability distributions

N. N. Chentsov

Moscow

Abstract: Let us consider a family of density functions (1) which describes a completely geodesic surface $\gamma$ in $\mathbf H(\Omega,S,I)$, see [l], , $s=(s^1,…,s^n)$ being a canonical affine parameter. Let us introduce the natural parameter $t=(t_1,…,t_n)$ defined by relation (4). It may be proved that the canonical parameter $s$ and the corresponding natural parameter $t$ are connected by the Legendre transformation and forme a couple of biorthogonal “coordinate systems”.
Theorem 2. A parameter $\theta$ of a smooth family of mutually absolutely continuous distributions admits a jointly efficient estimate if and only if the family is completely geodesic and $\theta$ is its natural parameter.
A probability distribution $P\in\mathbf H(\Omega,S,I)$ will be called a constructive distribution if there exists a mapping $\omega=\varphi(x)$ of $E[0\le x\le1]$ into $\Omega$ such that $P\{A\}=$the length of $\{x\colon\varphi(x)\in A\}$ for all $A\in S$. In accordance with  there is a system of constructive conditional distributions $P\{d\omega\mid\xi(\omega)=u\}$ on ($\Omega,S$) for every constructive $P$ and any random variable $\xi$. Let us now consider a category of families of constructive distributions with constructive Markov morfisms (20), and thus induced equivalence relation of families that is analogous to the one introduced in .
Theorem 3. Two completely geodesic families $\gamma_1$ and $\gamma_2$ of constructive probability distributions are statistically equivalent if and only if $\psi_1(s)=\psi_2(s)$ far some common canonical coordinatization $s$.
Theorem 4. If some canonical and some natural parametrizations of a constructive completely geodesic family $\gamma$ coinside then $\gamma$ is constructively equivalent to the family $\hat\gamma$ of normal distributions which have a common fixed matrix of variances, and the vector of means as their parameter.
Some examples are considered. Full text: PDF file (869 kB)

English version:
Theory of Probability and its Applications, 1966, 11:3, 425–435 Bibliographic databases:  Citation: N. N. Chentsov, “A systematic theory of exponential families of probability distributions”, Teor. Veroyatnost. i Primenen., 11:3 (1966), 483–494; Theory Probab. Appl., 11:3 (1966), 425–435 Citation in format AMSBIB
\Bibitem{Che66} \by N.~N.~Chentsov \paper A~systematic theory of exponential families of probability distributions \jour Teor. Veroyatnost. i Primenen. \yr 1966 \vol 11 \issue 3 \pages 483--494 \mathnet{http://mi.mathnet.ru/tvp642} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=203847} \zmath{https://zbmath.org/?q=an:0209.49704} \transl \jour Theory Probab. Appl. \yr 1966 \vol 11 \issue 3 \pages 425--435 \crossref{https://doi.org/10.1137/1111041} 

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