Suppose that $Y_1, Y_2, ..., Y_n$ is a random sample from a distribution with density function
$$ f(y) = \begin{cases} \theta y^{\theta - 1}\ \ \ \ 0 < y < 1, \\ 0\ \ \ \ \ \ \ \ \ \ \ elsewhere \end{cases} $$
The parameter $\theta$ is positive. Find the MVUE to estimate the parameter $\theta$ using the factorization criterion and Rao-Blackwell theorem.
I've been trying to solve it but without success.
The likelihood of the sample is
$$ L(y_1, y_2, ..., y_n | \theta) = \theta^n (\prod_{i=1}^n y_i)^{\theta - 1} $$
Since $L(\theta)$ is a function of $\theta$ and $\prod_{i=1}^n y_i$, then
$$ g(\prod_{i=1}^n y_i, \theta) = \theta^n (\prod_{i=1}^n y_i)^{\theta - 1} $$ and $$ h(y_1, y_2, ..., y_n) = 1 $$
By factorization criterion theorem, this implies that the estimator $U = \prod_{i=1}^n y_i$ is sufficient for the parameter $\theta$.
Then, when trying to use Rao-Blackwell theorem, I have that $E[U]=E[\prod_{i=1}^n y_i] = ...$ but I don't know how to solve this in order to get the factor to apply to U and make it unbiased.
Thanks for your time!
EDIT: So far, I have the sufficient estimator $U$. In order to use Rao-Blackwell, I need to solve $E[T|U]$ for some statistic $T$, so that would give the MVUE statistic $T^*$.
What would be the statistic $T$ in this case? And how could I solve the this conditional expectation?