Suppose that $X_1,X_2,...,X_n$ is a random sample from normal$(\mu, σ^2)$.
Find the UMVUE for $\mu ^{2}$ by assuming $\sigma ^{2}$ is unknown.
My approach:
The distribution of the sample mean, namely $$\bar X\sim\mathcal N\left(\mu,\frac{\sigma^2}{n}\right)$$
If $\sigma$ is known, a complete sufficient statistic for $\mu$ is $$\sum_{i=1}^n X_i \quad(\text{ and hence }\bar X)$$
Now,
\begin{align} \operatorname{Var}(\bar X)&=\frac{\sigma^2}{n} \\\implies E(\bar X^2)&=\frac{\sigma^2}{n}+\mu^2 \end{align}
That is, $$E\left(\bar X^2-\frac{\sigma^2}{n}\right)=\mu^2 $$
By Lehmann-Scheffe, $$\bar X^2-\frac{\sigma^2}{n}$$ is the UMVUE of $\mu^2$ when $\sigma^2$ is known.
My problem is how do I show the case where $\sigma^2$ is unknown?