Suppose that a statistic U has a probability density

Chapter 9, Problem 68E

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Problem 68E

Suppose that a statistic U has a probability density function that is positive over the interval a ≤ u ≤ b and suppose that the density depends on a parameter θ that can range over the interval α1 ≤ θ ≤ α2. Suppose also that g(u) is continuous for u in the interval [a, b]. If E[g(U ) | θ ] = 0 for all θ in the interval [α1, α2] implies that g(u) is identically zero, then the family of density functions { fU (u | θ), α1 ≤ θ ≤ α2} is said to be complete. (All statistics that we employed in Section 9.5 have complete families of density functions.) Suppose that U is a sufficient statistic for θ, and g1(U ) and g2(U ) are both unbiased estimators of θ. Show that, if the family of density functions for U is complete, g1(U ) must equal g2(U ), and thus there is a unique function of U that is an unbiased estimator of θ .

Coupled with the Rao–Blackwell theorem, the property of completeness of fU (u | θ), along with the sufficiency of U, assures us that there is a unique minimum-variance unbiased estimator (UMVUE) of θ .