7.7 The Case of Several Parameters

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Definition 7.7.1: Jointly sufficient statistic

Let

• $X_1,\dots ,X_n$ : Random sample, with
  • pdf/pmf $f(x;\mathbf{\theta })$, $\mathbf{\theta }\in \Omega \subset R^p$
• $\mathcal{S}$ : Support of $X$
• $\mathbf{Y}=(Y_1,\dots ,Y_m{)}^{\prime }$ : $m$-dimensional random vector of statistics, with
  • $Y_i=u_i(X_1,\dots ,X_n),i=1,\dots ,m$
  • pdf/pmf $f_{\mathbf{Y}}(\mathbf{y};\mathbf{\theta })$, $\mathbf{y}\in R^m$

Then $\mathbf{Y}$ is jointly sufficient for $\mathbf{\theta }$

If and only if

$$\frac{{\prod }_{i=1}^{n}f(x;\mathbf{\theta })}{f_{\mathbf{Y}}(\mathbf{y};\mathbf{\theta })}=H(x_1,\dots ,x_n),\forall x_i\in \mathcal{S}$$

Where $H(x_1,\dots ,x_n)$ does not depend upon $\mathbf{\theta }$

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Definition 7.7.2: Regular exponential class on random vectors

Let

• $X$ : Random variable, with
  • pdf/pmf $f(x;\mathbf{\theta })$, $\mathbf{\theta }\in \Omega \subset R^m$
• $\mathcal{S}$ : Support of $X$

Suppose

$$f(x;\mathbf{\theta })=\{\begin{array}{ll}\exp \left[{\sum }_{j=1}^m{p}_j(\mathbf{\theta })K_j(x)+H(x)+q(\mathbf{\theta })\right]& \forall x\in \mathcal{S}\\ 0& \text{elsewhere}\end{array}$$

Then we say $f(x;\mathbf{\theta })$ is a member of the exponential class

If

1. $\mathcal{S}$ does not depend upon $\mathbf{\theta }$
2. Space $\Omega$ contains a nonempty, $m$-dimensional open rectangle
3. $p_j(\mathbf{\theta })$ are all nontrivial, functionally independent, continuous functions of $\mathbf{\theta }$
4. If $X$ continuous random variable, then
   1. $K_j^{\prime }(x)$ are all continuous for $a<x<b$, not homogeneous linear function of the others.
   2. $H(x)$ : Continuous function of $x\in \mathcal{S}$
5. If $X$ discrete random variable, then
   1. $K_j^{\prime }(x)$ are all nontrivial functions of $x\in \mathcal{S}$, not homogeneous linear function of the others

Then we say $f(x;\mathbf{\theta })$ is a regular case of the exponential family

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Definition: Jointly complete sufficient statistics

Let

• $X_1,\dots ,X_n$ : Random sample on $X$ with pdf/pmf of $X$ is a regular case of the exponential class
• $Y_j={\sum }_{i=1}^n{K}_j(X_i),j=1,\dots ,m$ 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 ${\theta }_1,\dots ,{\theta }_m$

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