Let $X_1,...,X_n$ be a random sample from normal(θ,1). Is there an UMVUE of $θ^2$ here?
$X^2-1$ is an unbiased estimator of $θ^2$. First thing that came to my mind is to use Lehmann-Scheffe Theorem. But the problem is that, according to my calculations, n(θ,1) doesn't have a complete statistic since, even though $\overline{X}=\frac{\sum_{n=1}^{n}X_i}{n}$ is a minimal sufficient statistic for θ (easy to show), it is not complete (again easy to show). Thus there is simply no complete statistic for n(θ,1). Another way to prove it may use the condition that an UMVUE must be uncorrelated with all unbiased estimators of 0 (and vice-versa). But I don't have a candidate for UMVUE at all.
Can anybody give a clue about this question? Thanks!