What's the difference between Rao-Blackwell Theorem and Lehmann-Scheff Theorem?

Question

I know that the Rao-Blackwell theorem states that an unbiased estimator given a sufficient statistic will yield the best unbiased estimator. Is the only difference between Lehmann-Scheff and Rao-Blackwell that in Lehmann-Scheff, you need an unbiased estimator that is based on a complete sufficient statistic? I am also having a hard time conceptually understanding the definition of a complete statistic.

Answer 1

In point estimation, you could start by considering any unbiased estimator, say T, for a target parameter.

Then:

1. You could improve this estimator by getting E(T|S) where S is a sufficient statistic. That's theRao-Blackwell estimator.
2. Further, you could improve this by getting E(T|CS) where CS is a complete sufficient statistics. That's the based Lehman-Scheffe. This is the best unbiased estimator. You use typically use this method if you want to obtain the UMVUE (of course there are other ways of obtaining UMVUE).

Completeness simply means that the only unbiased estimator of zero is 0 itself.