Discussion
The following are some statements :
(I) If T is a boundedly complete sufficient statistic for \(\theta\) and S is an ancillary statistic for \(\theta\), then T and S are independent.
(II) An ancillary statistic may or may not be independent of a sufficient statistic.
(III) It is possible to have variance smaller than that of the Cramer-Rao lower bound at some points of the parameter space.
(IV) The parametric function need not be differentiable for the Cramer-Rao lower bound to hold.
Which of the following options is true ?
Answer: A
The true statements are (I), (II) and (III).
(I) is Basu's theorem: a boundedly complete sufficient statistic is independent of every ancillary statistic.
(II) is true because without completeness the independence can fail; when the sufficient statistic is not complete, an ancillary statistic can be a function of it.
(III) is true: the Cramer-Rao bound applies to unbiased estimators under regularity conditions. A biased estimator, such as a constant, has variance 0, below the bound, and when the regularity conditions fail (for example the uniform (0, θ) family) estimators can beat the bound.
(IV) is false: the bound \([\psi'(\theta)]^{2}/I(\theta)\) needs \(\psi\) to be differentiable. That rules out every option containing (IV).
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