Given $X_1,X_2,...,X_n$ random samples from $gamma(α,β)$ with unknown $α$. What will be the "Uniform minimum variance unbaised" Estimator for $α$ and $β$.
If $α$ is known then we know the UMVUE for $1/β$ is $(nα-1)/(\sum X_i)$. But can you please help with what if $α$ is unknown? i.e. what will be the estmator of $α$ and $β$.