In the setting of the EM algorithm, suppose that Y is the observed data and X is the complete data. Let Y and X have expected information matrices J(θ) and I(θ), respectively. Prove that I(θ)  J(θ)

in the notation of Problem 9.

If we could redesign our experiment so that X is observed directly, then invoke Problem 10 and argue that the standard error of the maximum likelihood estimate of any component θi will tend to decrease. (Hints: Using the notation of Section 2.4, let h(X | θ) = f(X | θ)/g(Y | θ) and prove that I(θ) − J(θ)=E{E[−d2 ln h(X | θ) | Y,θ]}.

The inner expectation on the right of this equation is an expected information.)