Does anyone know if there's a closed-form expression of the expectation of the maximum likelihood estimator of a binomial variable raised to a non-integer power?
Concretely, setting $$ P(m\mid N,p) = \binom{N}{m} p^m (1-p)^{N-m},\\ \hat{p} = \frac{m}{N}. $$
What is $E[\hat{p}^w]$, where $0<w<1$?
Specifically $$ E[\hat{p}^w] = \sum_{m=0}^N \binom{N}{m} p^m (1-p)^{N-m} \left(\frac{m}{N}\right)^w =~? $$
If $w$ was integer we could use the moment-generating function. Is there an analogous function that generates the "non-integer moments" of probability distributions?