Finding the Complete Sufficient Statistic

• January 21st 2010, 04:56 PM
statmajor
Finding the Complete Sufficient Statistic
Let $f(x; \theta) = \theta e^{-\theta x}$. Find the complete sufficient statistic for theta.

From what I understand, I have write the PDF in the "form" of the Regulat Exponential Class/Family which has the PDF of $f(x; \theta) = e^{p(\theta )K(x) + S(X) + q(x)}$

$f(x;\theta) = e^{-\theta x + log \theta}$

so I know $p( \theta) = - \theta$, $K(x) = x$ S(x) = 0 and $q( \theta ) = log \theta$

I don't understand my textbook's explaination of how to find the complete sufficient statistic, would someone please explain the next step I have to take?

Thank you.
• January 21st 2010, 06:05 PM
matheagle
Clearly $\sum X_i$ is suff for theta, next prove it is complete.
• January 21st 2010, 06:09 PM
statmajor
The only thing my book says is:

The statistic $Y_1 = \Sigma X_i$ is a sufficient statistic for theta and $f_{Y_1}(y_1; \theta)$ is complete, then so is Y1.

I have no idea what that means.