Exponential Family & Sufficient Statistic.

• Aug 19th 2009, 04:14 AM
panda*
Exponential Family & Sufficient Statistic.
The power family of distributions have densities

$f(x;\alpha,\theta) = \frac{\alpha x^{\alpha -1}}{\theta^{\alpha}}$, $0 \leq x \leq \theta$. 0 elsewhere.

If $\theta$ is known, show hat $f(x;\alpha)$ is in the exponential family and find a sufficient statistic for $\alpha$.
• Aug 19th 2009, 07:11 AM
matheagle
suff for $\alpha$ is $\Pi_{i=1}^n X_i$

and suff for $\theta$ is the largest $X_i$

both via the factorization theorem.
• Aug 19th 2009, 04:45 PM
panda*
Thank you, do you convert the pdf into the form of the exponential families to show? Then extract the term which is dependent on theta to determine the sufficient statistic?