A systematic theory of exponential families of probability distributions

Let us consider a family of density functions (1) which describes a completely geodesic surface $\gamma$ in $\mathbf H(\Omega,S,I)$, see [l], [4], $s=(s^1,\dots,s^n)$ being a canonical affine parameter. Let us introduce the natural parameter $t=(t_1,\dots,t_n)$ 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 [9] 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 [8].
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.