Multiparametric Sufficiency

Definition

Joint Sufficient Statistic

Let $X_1,X_2,\dots ,X_n$ be a Random Sample with PDF $f(x|\mathbf{\theta })$, where $\mathbf{\theta }\in \Omega \subset {\mathbb{R}}^p$, $\mathbf{Y}=(Y_1,Y_2,\dots ,Y_m{)}^{⊺}$, where $Y_i=u_i(X_1,X_2,\dots ,X_n)$ be a Statistic, and $f_{\mathbf{Y}}(\mathbf{y}|\mathbf{\theta })$ be a pdf of $\mathbf{y}$. The $\mathbf{Y}$ is jointly sufficient for $\mathbf{\theta }$ if and only $\frac{{\prod }_{i=1}^{n}f(x_i|\mathbf{\theta })}{f_{\mathbf{Y}}(\mathbf{y}|\mathbf{\theta })}=H(x_1,x_2,\dots ,x_n),\forall x_i\in \mathcal{S}$ where $H(x_1,x_2,\dots ,x_n)$ does not depend on $\mathbf{\theta }$

if and only if ${\prod }_{i=1}^{n}f(x_i|\mathbf{\theta })=k_1(\mathbf{y}|\mathbf{\theta })k_2(x_1,x_2,\dots ,x_n),\forall x_i\in \mathcal{S}$ where $k_2(x_1,x_2,\dots ,x_n)$ does not depend on $\mathbf{\theta }$

Completeness

Let $\{f(x_1,v_2,\dots ,v_k|\mathbf{\theta })|\mathbf{\theta }\in \Omega \}$ be a family of pdfs of $k$ random variables $V_1,V_2,\dots ,V_n$ If $\forall \mathbf{\theta }\in \Omega ,E[u(v_1,v_2,\dots ,v_n)]=0\Rightarrow u(v_1,v_2,\dots ,v_n)=0$, then the family of pdfs is called complete family, and $V_1,V_2,\dots ,V_k$ are complete statistics for $\mathbf{\theta }$

m-Dimensioanl Regular Exponential Class

Let $X$ be a Random Variable with PDF $f(x|\mathbf{\theta })$, where $\mathbf{\theta }\in \Omega \subset {\mathbb{R}}^m$, and $\mathcal{S}$ be a support of the $X$. If $f(x|\mathbf{\theta })$ can be written as $f(x|\mathbf{\theta })=\exp \left[\sum _{j=1}^m{p}_j(\mathbf{\theta })K_j(x)+H(x)+q({\theta }_1,{\theta }_2,\dots ,{\theta }_m)\right]I_{\mathcal{S}}(x)$ then, it is called $m$-dimensional exponential class. Further, it is called a regular case if the following are satisfied

• $\mathcal{S}$, support of $x$, does not depend on $\mathbf{\theta }$
• $\Omega$ contains m-dimensional open rectangle
• $p_1(\mathbf{\theta }),p_2(\mathbf{\theta }),p_m(\mathbf{\theta })$ are functionally independent and continuous function of $\mathbf{\theta }$
• If $X$ is a continuous random variable, then $K_j^{\prime }(x)$ and $H(x)$ are continuous function
• If $X$ is a discrete random variable, then $K_j(x)$ are non-trivial function of $x\in \mathcal{S}$

Let $X_1,X_2,\dots ,X_n$ be a Random Sample from a m-dimensional regular exponential class, then the joint pdf of $X_1,X_2,\dots ,X_n$ is ${\prod }_{i=1}^{n}f(x_i|\mathbf{\theta })=\exp \left[{\sum }_{j=1}^m{p}_j(\mathbf{\theta })\left({\sum }_{i=1}^n{K}_j(x_i)\right)+{\sum }_{i=1}^{n}H(x_i)+nq(\mathbf{\theta })\right]{\prod }_{i=1}^{n}I(x_i\in \mathcal{S})$

$\mathbf{Y}=(Y_1,Y_2,\dots ,Y_m{)}^{⊺}$, where $Y_j=\sum _{i=1}^n{K}_j(x_i)$, is a joint complete sufficient Statistic for $\mathbf{\theta }$

the joint pdf of $\mathbf{Y}=(Y_1,Y_2,\dots ,Y_m{)}^{⊺}$ is $f_{\mathbf{Y}}(\mathbf{y}|\mathbf{\theta })=R(\mathbf{y})\exp \left[\sum _{j=1}^m{p}_j(\mathbf{\theta })y_j+nq(\mathbf{\theta })\right]$, where $R(y)$ does not depend on $\mathbf{\theta }$

k-Dimensional Random Vector with p-Dimensional Parameters

Exponential Class

Let $\mathbf{X}$ be a $k$-dimensional random vector with pdf $f(\mathbf{x}|\mathbf{\theta })$, where $\mathbf{\theta }\in \Omega \subset {\mathbb{R}}^p$ $f(\mathbf{x}|\mathbf{\theta })=\exp \left[\sum _{j=1}^m{p}_j(\mathbf{\theta })K_j(\mathbf{x})+H(\mathbf{x})+q(\mathbf{\theta })\right]I_{\mathcal{S}}(\mathbf{x})$