# Math Help - Sufficiency and Maximum Likelihood.

1. ## Sufficiency and Maximum Likelihood.

Let $X_1, ... X_n$ be a random sample from the density function given by,

$f(x;\theta) = (\frac{1}{\theta})rx^{r-1}e^{-x^{r}/\theta}$

0 otherwise, $\theta > 0, x > 0$ for known r.

Find a sufficient statistic and maximum likelihood for $\theta$. Argue that the maximum likelihood estimator is also sufficient for $\theta$

2. By the factorization theorem

$\sum_{i=1}^n X_i^r$ or any multiple of it is suff for $\theta$

we get that from the likelihood function.

3. Then what's the maximum likelihood for theta? Is it possible to illustrate a few steps to kick start? Thank you!