Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 28J / b / iv / Solution

The assumed maximum-likelihood estimator is the sample mean of $h(X_i)$. Its central limit theorem isFor a regular model, the regular maximum-likelihood limit isThe two asymptotic variances must agree, soConsequentlyto first asymptotic order. The estimator therefore attains the Cramer-Rao bound asymptotically: its precision, the reciprocal variance, exhausts the information in the sample.