Finding the Complete Sufficient Statistic

**statmajor** · January 21st 2010, 03:56 PM

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.

**matheagle** · January 21st 2010, 05:05 PM

Clearly $\sum X_i$ is suff for theta, next prove it is complete.

**statmajor** · January 21st 2010, 05:09 PM

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.

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