Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 42 / 6 / viii

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 giveswhere $s_{\theta }={\partial }_{\theta }\log f_{\theta }$. 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 $j(\hat{\theta })/n$ 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 usesThe 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.