I am now studying complete sufficient statistic. My question is: Is there any relationship between the existence of complete sufficient statistic and the existence of unbiased estimator?
I know that by Lehmann-Scheffe Theorem, if an unbiased estimator of $\theta$(parameter) exists as a function of a complete sufficient statistic, then it should be unique up to a.s. sense. What I am wondering is: Can we always find such an unbiased estimator if we have complete sufficient statistic? Or conversely, if an unbiased estimator of $\theta$(parameter) exists as a function of a sufficient statistic, does that imply that the sufficient statistic is complete?
I tried to come up with some examples but failed to think of any. For example, is there an example s.t. a complete sufficient statistic exists whereas an unbiased estimator does not exists as a function of the complete sufficient statistic? I'd really appreciate your comment. Thanks!
Best Answer
A slight modification of the one given in the comments. Let $X_1,X_2,...X_n$ follow $B(m,\theta)$. Then the function $g(\theta)=\frac{1}{\theta}$ doesn't admit an unbiased estimator while $\sum_{i=1}^{n}{X_i}$ is a Complete Sufficient Statistic.
Let $X_1,X_2,...X_n$ follow $P(\theta)$ then $S^2 = \frac{1}{n-1}\sum_{i=1}^{n}{{(X_i-\bar{X}})}^2$ is an unbiased estimator and is a function of a statistic $T(X) = (\sum_{i=1}^{n}{X_i},\sum_{i=1}^{n}{X_i}^2)$ which is sufficient. But $S^2$ is not a Complete Sufficient Statistic.