Jointly Sufficient Statistic

Definition

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 $\mathbf{\theta }$