= Wald and likelihood-ratio asymptotic equivalence
{c}
In a regular scalar model under the null, Taylor expansion about the MLE gives
$$
2\{\ell_n(\widehat\theta)-\ell_n(\theta_0)\}
=-\ell_n''(\widetilde\theta)(\widehat\theta-\theta_0)^2.
$$
A uniform law of large numbers for the observed information makes
$-\ell_n''(\widetilde\theta)/n$ and the Fisher information used in the Wald statistic converge to the same positive limit. Their ratio therefore converges in probability to one.
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