7.7 The Case of Several Parameters

Definition 7.7.1: Jointly sufficient statistic

Let

$X_{1}, \dots, X_{n}$ : Random sample, with
- pdf/pmf $f (x; θ)$ , $θ \in Ω \subset R^{p}$
$S$ : Support of $X$
$Y = (Y_{1}, \dots, Y_{m})^{'}$ : $m$ -dimensional random vector of statistics, with
- $Y_{i} = u_{i} (X_{1}, \dots, X_{n}), i = 1, \dots, m$
- pdf/pmf $f_{Y} (y; θ)$ , $y \in R^{m}$

Then $Y$ is jointly sufficient for $θ$

If and only if

\frac{\prod _{i = 1}^{n} f ( x ; θ )}{f _{Y} ( y ; θ )} = H (x_{1}, \dots, x_{n}), \forall x_{i} \in S

Where $H (x_{1}, \dots, x_{n})$ does not depend upon $θ$

Let

Suppose

f (x; θ) = {exp [\sum_{j = 1}^{m} p_{j} (θ) K_{j} (x) + H (x) + q (θ)] 0 \forall x \in S elsewhere

Then we say $f (x; θ)$ is a member of the exponential class

$S$ does not depend upon $θ$
Space $Ω$ contains a nonempty, $m$ -dimensional open rectangle
$p_{j} (θ)$ are all nontrivial, functionally independent, continuous functions of $θ$
If $X$ continuous random variable, then
1. $K_{j}^{'} (x)$ are all continuous for $a < x < b$ , not homogeneous linear function of the others.
2. $H (x)$ : Continuous function of $x \in S$
If $X$ discrete random variable, then
1. $K_{j}^{'} (x)$ are all nontrivial functions of $x \in S$ , not homogeneous linear function of the others

Then we say $f (x; θ)$ is a regular case of the exponential family

Let

$X_{1}, \dots, X_{n}$ : Random sample on $X$ with pdf/pmf of $X$ is a regular case of the exponential class
$Y_j = \sum_{i=1}^n K_j(X_i), \quad j=1, \dots, m $$ [[3 Reference/def-jointly-sufficient-statistic_202507171021\|jointly sufficient statistics]] for parameters$ \theta_{1},\dots,\theta_{m}$

If $n > m$

Then

The family of pdfs of these $Y_{1}, Y_{2}, \dots, Y_{m}$ is complete.
We refer to $Y_{1}, Y_{2}, \dots, Y_{m}$ as joint complete sufficient statistics for $θ_{1}, \dots, θ_{m}$