Consider $X_1,X_2,...X_n$ iid random sample from exponential distribution with pdf: $$f(x,\theta)=\theta e^{-\theta x},\: x>0$$
I found that, $$E\left[\frac{n-1}{n\bar{X}}\right]=\theta$$
So $$\hat{\theta}=\frac{n-1}{n\bar{X}}$$ is an Unbiased estimator of $\theta$.
Also the Fisher information is $I(\theta)=\frac{1}{\theta^2}$
The Cramer Rao Lower bound is $$Var(\hat{\theta})\ge \frac{\theta^2}{n}$$
Using the fact that $n\bar{X} \sim \Gamma(n, \theta)$
I found that $$Var(\hat{\theta})=\frac{\theta^2}{n-2}$$
So since the variance is not exactly CRLB, it is not efficient and hence not UMVUE?
But the solution here says it is UMVUE. Finding UMVUE of $\theta$ when the underlying distribution is exponential distribution
