Thread: Exponential Family & Sufficient Statistic.

1. 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 $\alpha$.

2. suff for $\alpha$ is $\Pi_{i=1}^n X_i$

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

both via the factorization theorem.

3. 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?