$\newcommand{\E}{\mathbb{E}}$ $\newcommand{\V}{\mathbb{V}}$ Let $p\in\mathbb{N}$. Is there a nice general expression for the UMVUE of $\mu^p$, where $X_1,\cdots,X_n\sim\mathcal{N}(\mu,\sigma^2)$ are i.i.d.?
In the case of $p=2$, we know that $$\E[\bar{X}^2]=\sigma^2/n+\mu^2,$$so $$\begin{align}\mu^2&=\E[\bar{X}^2]-\sigma^2/n\\&=\E[\bar{X}^2-S^2/n].\end{align}$$Since $(\bar{X},S^2)$ is a complete sufficient statistic for $(\mu,\sigma^2)$, by the above and Lehmann-Scheffe' $T=\bar{X}^2-S^2/n$ must be the UMVUE of $\mu^2$. In general it looks like one can always write $$\mu^p=\E[\text{some polynomial in }\bar{X},S^2]$$but I'm not 100% sure on the specifics. For $p=3$ its $$T=\bar{X}^3-\frac{3\bar{X}S^2}{n}.$$ This was inspired by Casella-Berger exercises 7.47, 7.59 and is not homework.
