Does linearity hold for UMVUE? Based on my intuition, if $T$ is the UMVUE of $g($$\theta$$)$, then $aT+b$ is the UMVUE of $ag($$\theta$$)+b$ ($a$ and $b$ are constants).
I tried to prove it using the CR inequality of $g($$\theta$$)$ as follows:
$$Var(T) \geq \frac{[\frac{d}{d\theta} g(\theta)]^2}{I_{X_1,...,X_n}(\theta)}$$
But $$[\frac{d}{d\theta} ag(\theta)+b]^2 = a^2[\frac{d}{d\theta} g(\theta)]^2$$ and $$a^2Var(T) = Var(aT) = Var(aT+b),$$ so the UMVUE of $ag($$\theta$$)+b$ is $aT+c$, where $c$ is a constant that can take any value.
Did I prove it the wrong way?
 A: Since $T$ is UMVUE of its expectation, it is (trivially) assumed that first and second moments of $T$ are finite. This in turn  implies the first two moments of $aT+b$ are also finite. 
Clearly $aT+b$ is unbiased for $ag(\theta)+b$. To show this estimator has the minimum variance among the class of all unbiased estimators, you would need to use a result like this one which says that a necessary and sufficient condition for an unbiased estimator (with finite second moment) to be UMVUE of some parametric function is that it must be uncorrelated with every unbiased estimator of zero.
So suppose $\mathcal U_0$ is the class of all unbiased estimators of zero with finite variances. And $\Omega$ is the set of admissible values of $\theta$.
Since $T$ is UMVUE of its expectation, by the result above you have
$$\operatorname{Cov}_{\theta}(T,h)=\operatorname E_{\theta}(Th)=0\quad\forall\,\theta\in\Omega,\,\forall \,h\in \mathcal U_0$$
Therefore,
$$\operatorname E_{\theta}\left[(aT+b)h\right]=a\operatorname E_{\theta}(Th)+b\operatorname E_{\theta}(h)=0\quad\forall\,\theta\in\Omega,\,\forall \,h\in \mathcal U_0$$
That is,
$$\operatorname{Cov}_{\theta}(aT+b,h)=0\quad\forall\,\theta\in\Omega,\,\forall \,h\in \mathcal U_0$$
So $aT+b$ is uncorrelated with every unbiased estimator of zero, proving that it is UMVUE of its expectation $ag(\theta)+b$.
As you might expect, one can extend this argument to say that any real polynomial in $T$ would be the UMVUE of its expectation. Or even a linear combination of UMVUEs is also the UMVUE of its expectation.
