In a Gaussian shift, tests and confidence sets are projections and ellipsoids in the information metric. The Wald, score, and likelihood-ratio constructions are three coordinate descriptions of the same local normal component.
Smooth null hypotheses
Let \(\theta\in\mathbb R^p\) and consider \[ H_0:g(\theta)=0, \] where \(g:\mathbb R^p\to\mathbb R^q\) has derivative \[ G=\dot g_{\theta_0} \] of full row rank \(q\). The local null tangent space is \[ T_0=\{h:Gh=0\}. \] Its information-orthogonal complement contains the \(q\) directions tested by the hypothesis.
The Wald statistic
Definition. If \[ \sqrt n(\hat\theta_n-\theta_0) \rightsquigarrow N_p(0,I^{-1}), \] the Wald statistic is \[ W_n = n\,g(\hat\theta_n)^\top \{G I^{-1}G^\top\}^{-1} g(\hat\theta_n), \] with consistent plug-in estimates used in practice.
Proposition. Under \(H_0\), \[ W_n\rightsquigarrow\chi_q^2. \]
Proof. The delta method gives \[ \sqrt n\,g(\hat\theta_n) = G\sqrt n(\hat\theta_n-\theta_0)+o_p(1) \rightsquigarrow N_q(0,GI^{-1}G^\top). \] Standardizing this Gaussian vector and taking its squared norm yields \(\chi_q^2\). \(\square\)
Score and likelihood-ratio statistics
Theorem. Let \(\hat\theta_{0,n}\) be the constrained MLE. Under uniform LAQ and regularity, \[ LR_n = 2\{\ell_n(\hat\theta_n)-\ell_n(\hat\theta_{0,n})\}, \] the efficient score statistic \(S_n\), and the Wald statistic satisfy \[ LR_n=S_n+o_p(1)=W_n+o_p(1) \rightsquigarrow\chi_q^2. \]
Proof. In local coordinates the unconstrained quadratic maximizer is \(\hat h=I^{-1}\Delta\), and the null maximizer is its \(I\)-orthogonal projection onto \(T_0\). The likelihood-ratio statistic is the squared information distance of \(\hat h\) from \(T_0\). The efficient score removes the score component tangent to \(T_0\) and takes the same squared norm. The Wald statistic measures the corresponding normal displacement through \(G\). These are algebraically identical quadratic forms in the Gaussian central sequence, up to estimated-information and likelihood remainders. \(\square\)
Local alternatives and noncentrality
Theorem. Under \[ \theta_n=\theta_0+h/\sqrt n, \] the three statistics converge to \[ \chi_q^2(\lambda), \qquad \lambda = (Gh)^\top (GI^{-1}G^\top)^{-1} Gh. \]
The noncentrality is the squared information distance of the local alternative from the null tangent space. A level-\(\alpha\) test has limiting power \[ P\{\chi_q^2(\lambda)>\chi_{q,1-\alpha}^2\}. \]
Confidence ellipsoids
Definition. An asymptotic \(1-\alpha\) confidence ellipsoid is \[ C_n = \left\{ \theta: n(\hat\theta_n-\theta)^\top \hat I_n(\hat\theta_n-\theta) \leq\chi_{p,1-\alpha}^2 \right\}. \]
Proposition. If the asymptotic normality and information consistency hold uniformly on compact parameter sets, then \[ \inf_{\theta\in K}P_\theta(\theta\in C_n)\to1-\alpha. \]
Pointwise normality gives only pointwise coverage. Uniform coverage requires uniform control of the estimator approximation and information estimate.
Test–confidence duality
Proposition. Let \(\varphi_{\theta,n}\) be a level-\(\alpha\) test of the point null \(H_0:\theta=\theta_0\). Then \[ C_n(X)=\{\theta:\varphi_{\theta,n}(X)=0\} \] is a \(1-\alpha\) confidence set. Conversely, every confidence set defines tests by rejection when \(\theta\notin C_n\).
One-sided constraints
If the null lies on a boundary or is defined by inequalities, the local parameter set is a cone rather than a linear subspace. Projection of a Gaussian vector onto a cone yields a chi-bar-square distribution, a mixture of chi-square laws with different degrees of freedom. Wilks' ordinary \(\chi_q^2\) conclusion therefore fails at boundaries.
Weak identification and growing dimension
Wald, score, and likelihood-ratio equivalence assumes nonsingular information and fixed dimension. Near-singular information, nuisance parameters unidentified under the null, or \(p=p_n\to\infty\) can produce different limits. The Gaussian experiment must be reidentified at the correct scale before using standard ellipsoids.