I am having some trouble with the following problem:
Let $X = (X_{1}, . . . , X_{n})$ a random sample from $f_{\theta}$, where $\theta \in \Theta$. Suppose that $W$ is the MVUE for $\theta$. Let $Z$ be an unbiased estimator for the single point $0$; that is, for some function $u$ that does not depend on $\theta$, we have $Z = u(X)$ and $\mathbb{E} \theta Z = 0$ for all $\theta$. Show that $Cov_{\theta} (W,Z) = 0$.
I know that this problem is related to completeness but I am not sure what do I need to do in order to prove it. Do I need to apply the Lehmann-Scheffe Theorem?
Thanks for any help!