Suppose that X1, . . . , Xn form a random sample from a distribution for which the p.d.f. or the p.f. is f (x|θ), where the value of the parameter θ is unknown. Let X = (X1, . . . , Xn), and let T be a statistic. Assume that δ(X) is an unbiased estimator of θ such that Eθ [δ(X)|T ] does not depend on θ. (If T is a sufficient statistic, as defined in Sec. 7.7, then this will be true for every estimator δ. The condition also holds in other examples.) Let δ0(T ) denote the conditional mean of δ(X) given T .
a. Show that δ0(T) is also an unbiased estimator of θ.
b. Show that Varθ (δ0) ≤ Varθ (δ) for every possible value of θ.