Joint Complete Sufficient Statistics

Definition

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 ${\theta }_1,\dots ,{\theta }_m$