= Solution
For a regular iid model with a fixed-dimensional interior parameter $\theta_0$, identifiability, sufficient differentiability and moment domination, and nonsingular per-observation <Fisher information> $I(\theta_0)$, a consistent <maximum-likelihood estimator> solves the <score equation>. Expanding it gives
$$
\sqrt n(\widehat\theta-\theta_0)=I(\theta_0)^{-1}\frac1{\sqrt n}\sum_i s_{\theta_0}(Y_i)+o_p(1)
\Rightarrow N_p(0,I(\theta_0)^{-1}),
$$
where $s_\theta=\partial_\theta\log f_\theta$. The score has mean zero and <covariance> $I$ under the regularity assumptions. This proves the usual large-sample information bound and asymptotic efficiency within regular parametric inference. The observed curvature $j(\widehat\theta)/n$ and the empirical score <covariance> consistently estimate $I$ when the model is correctly specified.
For $r$ smooth constraints $h(\theta)=0$ with full-row-rank derivative $H$, the <Wald test> uses
$$
W=n\,h(\widehat\theta)^T[H I^{-1}H^T]^{-1}h(\widehat\theta)\Rightarrow\chi_r^2.
$$
The <likelihood-ratio test> and <score test>, evaluated using the constrained fit and nuisance-adjusted efficient information, have the same first-order chi-square law. A quadratic local <likelihood> expansion explains their asymptotic equivalence. <Profile likelihood> supplies inference about a parameter of interest with fitted <nuisance parameters>, and the corresponding chi-square cutoffs give approximate confidence regions.
These conclusions depend on the stated regularity: parameters at a boundary, nonidentifiability, singular information, changing dimension, or parameter-dependent support can give different rates and limits. Under misspecification, the fitted parameter generally approaches a pseudo-true value and its <variance> is a <sandwich covariance matrix>, not simply inverse <Fisher information>.
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