Wald, score, and likelihood-ratio asymptotic chi-square calibration

If $d=q$, then $I_{\mathrm{e}\mathrm{f}\mathrm{f}}=I$ is positive definite. If $d>q$, $I_{\eta \eta }$ is a positive-definite principal block. For nonzero $v\in {\mathbb{R}}^q$, evaluate the quadratic form of $I$ at $(v,-I_{\eta \eta }^{-1}{I}_{\eta \psi }v)$; the result is $v^T{I}_{\mathrm{e}\mathrm{f}\mathrm{f}}v>0$, so $I_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is positive definite in either case. Normalizing $p_{{\theta }_0+h}$ in the QMD expansion [F1], write $\sqrt{p_{{\theta }_0+h}}=\sqrt{p_{{\theta }_0}}+\frac{1}{2}{h}^T{s}_{{\theta }_0}\sqrt{p_{{\theta }_0}}+e_h$ with $\mathrm{\parallel }{e}_h{\mathrm{\parallel }}_{L^2(\mu )}=o(\mathrm{\parallel }h\mathrm{\parallel })$. Expanding the squared norm, whose value is $\int p_{{\theta }_0+h}d\mu =1$, gives $h^T{\mathbb{E}}_{{\theta }_0}{s}_{{\theta }_0}+O(\mathrm{\parallel }h{\mathrm{\parallel }}^2)+o(\mathrm{\parallel }h\mathrm{\parallel })=0$. Taking $h=t{e}_j$ and dividing by $t$ proves each score coordinate has mean zero. Since $s_{{\theta }_0}$ is square integrable by [F1], the iid CLT [F7] gives ${\mathrm{\Delta }}_n\Rightarrow N_d(0,I)$ under AC. Define $r_n={\mathrm{\Delta }}_{\psi ,n}-I_{\psi \eta }{I}_{\eta \eta }^{-1}{\mathrm{\Delta }}_{\eta ,n}$. Its covariance, by block multiplication, is $\mathrm{Var}(r_n)=I_{\psi \psi }-I_{\psi \eta }{I}_{\eta \eta }^{-1}{I}_{\eta \psi }-I_{\psi \eta }{I}_{\eta \eta }^{-1}{I}_{\eta \psi }+I_{\psi \eta }{I}_{\eta \eta }^{-1}{I}_{\eta \eta }{I}_{\eta \eta }^{-1}{I}_{\eta \psi }=I_{\mathrm{e}\mathrm{f}\mathrm{f}}\mathrm{.}$ Thus the linear-map case of Slutsky [F6] gives $r_n\Rightarrow N_q(0,I_{\mathrm{e}\mathrm{f}\mathrm{f}})$; its covariance is the block calculation above.

By [F3], the full MLE satisfies $\sqrt{n}({\widehat{\theta }}_n-{\theta }_0)=I^{-1}{\mathrm{\Delta }}_n+o_P(1)$. Solving the block system $I(a,b{)}^T=({\mathrm{\Delta }}_{\psi ,n},{\mathrm{\Delta }}_{\eta ,n}{)}^T$ shows its tested block is $a=I_{\mathrm{e}\mathrm{f}\mathrm{f}}^{-1}{r}_n$, including the empty-nuisance case. If $d=q$, the local null contains only ${\theta }_0$, so consistency makes the constrained MLE equal to ${\theta }_0$ with probability tending to one and there is no nuisance expansion to prove. Otherwise, [F3] makes the constrained root-$n$ error bounded. In local coordinates $h=(u,v)$ for $(\eta ,\psi )$, the limiting quadratic log likelihood is $Q_n(u,v)=u^T{\mathrm{\Delta }}_{\eta ,n}+v^T{\mathrm{\Delta }}_{\psi ,n}-\frac{1}{2}\{u^T{I}_{\eta \eta }u+2u^T{I}_{\eta \psi }v+v^T{I}_{\psi \psi }v\}\mathrm{.}$ For general $v$, completing the square in $u$ gives the unique maximizer $u=I_{\eta \eta }^{-1}({\mathrm{\Delta }}_{\eta ,n}-I_{\eta \psi }v)$ and the profile $v^T{r}_n-\frac{1}{2}{v}^T{I}_{\mathrm{e}\mathrm{f}\mathrm{f}}v$ up to a term independent of $v$. At the null $v=0$ this gives the unique nuisance maximizer $u_0=I_{\eta \eta }^{-1}{\mathrm{\Delta }}_{\eta ,n}$. On every distance-$\epsilon$ set from $u_0$, strict positive definiteness of $I_{\eta \eta }$ makes the quadratic log likelihood at least $\frac{1}{2}{\lambda }_{\min }(I_{\eta \eta }){\epsilon }^2$ below its maximum. Uniformity of the likelihood expansion [A1], constrained root-$n$ tightness [F3], and this gap therefore give $\sqrt{n}({\tilde{\eta }}_n-{\eta }_0)=I_{\eta \eta }^{-1}{\mathrm{\Delta }}_{\eta ,n}+o_P(1)$.

Consistency places both maximizers in the open chart neighborhood with probability tending to one; the off-chart zero conventions in [F2] therefore do not affect the expansions below. The upper-left block of $I^{-1}$ is $I_{\mathrm{e}\mathrm{f}\mathrm{f}}^{-1}$ by solving the block system $I(x,y{)}^T=(a,0{)}^T$. The differentiable local chart has tested coordinate $\psi$, so the Wald statistic in [F2], the expansion in step 2.1, and covariance consistency give $W_n=(I_{\mathrm{e}\mathrm{f}\mathrm{f}}^{-1}{r}_n{)}^T{I}_{\mathrm{e}\mathrm{f}\mathrm{f}}(I_{\mathrm{e}\mathrm{f}\mathrm{f}}^{-1}{r}_n)+o_P(1)=r_n^T{I}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{-1}{r}_n+o_P(1)\mathrm{.}$

If $d=q$, the nuisance score is empty and the score expansion [A1] at $h=0$ gives $U_{\mathrm{e}\mathrm{f}\mathrm{f},n}=U_{\psi ,n}({\theta }_0)={\mathrm{\Delta }}_n+o_P(1)=r_n+o_P(1)$. If $d>q$, the score expansion [A1] at the constrained MLE and $\sqrt{n}({\tilde{\eta }}_n-{\eta }_0)=I_{\eta \eta }^{-1}{\mathrm{\Delta }}_{\eta ,n}+o_P(1)$ give $U_{\psi ,n}({\tilde{\theta }}_n)=r_n+o_P(1)$ and $U_{\eta ,n}({\tilde{\theta }}_n)=o_P(1)$. The estimated projection coefficient is bounded in probability by consistency, so the efficient score in [F2] is $r_n+o_P(1)$. Efficient-information consistency and [F5] imply $S_n=r_n^T{I}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{-1}{r}_n+o_P(1)$.

The likelihood expansion in [A1] makes $L_n$ equal, up to $o_P(1)$, twice the maximum of $Q_n(u,v)$ over all local $(u,v)$ minus twice its maximum over the null $v=0$. The unrestricted and restricted quadratic maximizers are linear functions of ${\mathrm{\Delta }}_n$, hence are $O_P(1)$ by step 1.1 and [F5]. Taking a bounded local ball large enough with probability arbitrarily close to one justifies using the uniform expansions for both maxima. The profile formula in step 2.1 reduces this difference to $2{\sup }_v\{v^T{r}_n-\frac{1}{2}{v}^T{I}_{\mathrm{e}\mathrm{f}\mathrm{f}}v\}=r_n^T{I}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{-1}{r}_n\mathrm{.}$ The identity follows by completing the square at $v=I_{\mathrm{e}\mathrm{f}\mathrm{f}}^{-1}{r}_n$. Hence $L_n=r_n^T{I}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{-1}{r}_n+o_P(1)$.

Since $r_n\Rightarrow N_q(0,I_{\mathrm{e}\mathrm{f}\mathrm{f}})$ and $I_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is positive definite, its transformed vector $I_{\mathrm{e}\mathrm{f}\mathrm{f}}^{-1\mathrm{/}2}{r}_n$ converges to $N_q(0,I_q)$. The squared Euclidean norm of that limit is ${\chi }_q^2$ by [F9]. Combining this with steps 3.1–3.3 and [F4]–[F5] proves all three limits and their common first-order approximation. AC is inherited through [F2], [F3], and [F7]–[F9], as recorded in [A3].