Quadratic likelihood approximation is the operational form of LAN. Completing the square produces a local Gaussian center, one Newton step, likelihood-ratio statistics, and information-metric projections under constraints.
Local asymptotic quadraticity
Definition. Around a center \(\tilde\theta_n\), with local coordinate \(u=r_n(\theta-\tilde\theta_n)\), a local asymptotic quadratic expansion is \[ \ell_n(\tilde\theta_n+r_n^{-1}u)-\ell_n(\tilde\theta_n) = u^\top S_n-\frac12u^\top J_nu+r_n(u), \] where \(J_n\) is positive definite and \[ \sup_{\|u\|\leq C}|r_n(u)|\to_p0 \] for every fixed \(C\).
Proposition. The quadratic part satisfies \[ u^\top S_n-\frac12u^\top J_nu = -\frac12(u-\hat u_n)^\top J_n(u-\hat u_n) +\frac12S_n^\top J_n^{-1}S_n, \] where \[ \hat u_n=J_n^{-1}S_n. \]
Proof. Expand the right side: \[ -\frac12u^\top J_nu +u^\top J_n\hat u_n -\frac12\hat u_n^\top J_n\hat u_n +\frac12S_n^\top J_n^{-1}S_n. \] Since \(J_n\hat u_n=S_n\), the final two terms cancel and the expression reduces to the left side. \(\square\)
One-step estimation
Theorem. Suppose the LAQ remainder is uniform on bounded sets, \(J_n\to_pJ\succ0\), and the exact local maximizer is tight. Then \[ r_n(\hat\theta_n-\tilde\theta_n) = J_n^{-1}S_n+o_p(1). \] The one-step estimator \[ \theta_n^{(1)} = \tilde\theta_n+r_n^{-1}J_n^{-1}S_n \] is asymptotically equivalent to the local maximum-likelihood estimator.
Proof. The quadratic criterion has a unique maximizer \(\hat u_n=J_n^{-1}S_n\). Uniform \(o_p(1)\) perturbation on every bounded set and tightness of both maximizers imply, by the argmax theorem, \[ r_n(\hat\theta_n-\tilde\theta_n)-\hat u_n=o_p(1). \] The one-step estimator uses exactly \(\hat u_n\). \(\square\)
Newton and Fisher scoring forms
For a twice differentiable log likelihood, \[ S_n=r_n^{-1}\dot\ell_n(\tilde\theta_n), \qquad J_n=r_n^{-1} \{-\ddot\ell_n(\tilde\theta_n)\} r_n^{-1}. \] With \(r_n=\sqrt n I_p\), the update becomes \[ \theta_n^{(1)} =\tilde\theta_n+ \{-\ddot\ell_n(\tilde\theta_n)\}^{-1} \dot\ell_n(\tilde\theta_n), \] the Newton step. Replacing observed information by expected Fisher information gives Fisher scoring.
Likelihood-ratio gain
Corollary. The increase from the local center to the unconstrained quadratic maximum is \[ 2\{\ell_n(\hat\theta_n)-\ell_n(\tilde\theta_n)\} = S_n^\top J_n^{-1}S_n+o_p(1). \]
If \(S_n\rightsquigarrow N(0,J)\), this quadratic form converges to \(\chi_p^2\).
Constrained quadratic maximization
Proposition. Let the local null tangent space be a linear subspace \(T\subset\mathbb R^p\). The constrained maximizer is the \(J_n\)-orthogonal projection of \(\hat u_n=J_n^{-1}S_n\) onto \(T\): \[ \hat u_{0,n}=\Pi_T^{J_n}\hat u_n. \] The likelihood-ratio statistic is \[ \|\hat u_n-\Pi_T^{J_n}\hat u_n\|_{J_n}^2+o_p(1). \]
Proof. Completing the square shows that maximizing the quadratic over \(T\) is equivalent to minimizing \(\|u-\hat u_n\|_{J_n}^2\) over \(u\in T\). The difference between unrestricted and constrained maxima is one half the squared distance; multiplying by two gives the statistic. \(\square\)
Observed and expected information
Observed information adapts to the realized curvature and is natural for Newton optimization. Expected information defines the deterministic LAN geometry. Under regularity, \[ -n^{-1}\ddot\ell_n(\theta_0)\to_p I(\theta_0), \] so the two are first-order equivalent. In small samples, near boundaries, or weakly identified models, their difference can be substantial and the quadratic approximation may be unreliable.
Remainder requirements
A Taylor formula alone is not an asymptotic theorem. One must show the third-order or stochastic remainder is uniform over the local region containing the maximizer. If dimension grows with \(n\), the norm of \(u\) and operator fluctuations of \(J_n\) can make a fixed-dimensional \(o_p(1)\) argument fail.