Suppose we are testing $\mathbf h(\boldsymbol\theta) = \mathbf 0$ versus $\mathbf h(\boldsymbol\theta) \neq \mathbf 0$ for a vector of parameters $\boldsymbol\theta \in \boldsymbol\Theta\subset \mathbb R^k $, and $\mathbf h:\mathbb R^k \to \mathbb R^q$. If we can calculate the log-likelihood of our data $\ell(\boldsymbol\theta \mid \mathbf x)$ given observed data $\mathbf x$, then we can test our hypothesis with the generalized likelihood ratio: $$ LR = -2\left[\ell(\hat{\boldsymbol\theta}_\text{RMLE} \mid \mathbf x)-\ell(\hat{\boldsymbol\theta}_\text{MLE} \mid \mathbf x)\right].$$ The estimator $\hat{\boldsymbol\theta}_\text{RMLE}$ is the MLE estimate subject to the constraint $\mathbf h(\boldsymbol\theta)$. I'm trying to show that $LR\overset{d}{\to}\chi^2_{q-k}$, but struggling to do so in full generality considering $\mathbf h$ may be nonlinear. In this case Engel (1984) argues that it doesn't really matter if $\mathbf h$ is nonlinear, because we can approximate it linearly using it's Jacobian (assuming $\mathbf h$ is differentiable). I'm a bit confused by his argument, particular when he says "for the null and local alternatives $\theta$ approaches $\theta_0$." I appreciate any pointers or advice!



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