= Solution
The assumed <maximum-likelihood estimator> is the sample mean of $h(X_i)$. Its <central limit theorem> is
$$
\sqrt n(\widehat\theta_{\mathrm{MLE}}-\theta_0)
\xrightarrow d N\left(0,\operatorname{Var}_{\theta_0}(h(X))\right).
$$
For a regular model, the regular maximum-likelihood limit is
$$
\sqrt n(\widehat\theta_{\mathrm{MLE}}-\theta_0)
\xrightarrow d N(0,I(\theta_0)^{-1}).
$$
The two asymptotic variances must agree, so
$$
\boxed{I(\theta_0)=\frac1{\operatorname{Var}_{\theta_0}(h(X))}.}
$$
Consequently
$$
\operatorname{Var}(\widehat\theta_{\mathrm{MLE}})
=\frac{\operatorname{Var}(h(X))}{n}
=\frac1{nI(\theta_0)}
$$
to first asymptotic order. The estimator therefore attains the <Cramér-Rao bound> asymptotically: its precision, the reciprocal variance, exhausts the information in the sample.
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