Complete Sufficient Statistic

Definition

Let $X_1,\dots ,X_n$ : Random sample, with pdf/pmf $f(x;\theta )$, $\theta \in \Omega$

If

• $Y_1=u_1(X_1,\dots ,X_n)$ : Sufficient statistic for $\theta$ with pdf/pmf $f_{Y_1}(y_1;\theta )$
• Family $\{f_{Y_1}(y_1;\theta ):\theta \in \Omega \}$ be complete

Then $Y_1$ is a complete sufficient statistic for $\theta$

Remark

The statement ”$Y_1$ is a sufficient statistic for a parameter $\theta$, $\theta \in \Omega$, and that the family $\{f_{Y_1}(y_1;\theta ):\theta \in \Omega \}$ of probability density functions is complete” is lengthy and somewhat awkward.

We shall adopt the less descriptive, but more convenient, terminology that $Y_1$ is a complete sufficient statistic for $\theta$.

Related theorems

• Theorem 7.5.2