3
$\begingroup$

If a statistic is complete, then any function of that statistic is complete. Is it true in general? For example, I know that for a gamma distribution $G(2,\theta)$, Sample mean will be a complete statistic. Then, can I say that any function of sample mean will be a complete statistic?

$\endgroup$
4
$\begingroup$

If $T$ is complete and $V=f(T)$, then, if $$\mathbb{E}_\theta[\Psi(V)]=c$$ for all $\theta$'s, this implies that $$\mathbb{E}_\theta[\Psi(f(T))]=\mathbb{E}_\theta[(\Psi\circ f)(T)]=c$$ for all $\theta$'s. Therefore, $\Psi\circ f$ is constant on the support of $T$ (i.e., the set of the possible values taken by $T$), $$(\Psi\circ f)(t)=\Psi(f(t))=c\qquad t\in\text{supp}\,T(\mathcal{X})$$hence$$\Psi(v)=c\qquad v\in\text{supp}\,fT(\mathcal{X}))$$which shows $V$ is complete.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.