# Asymptotic Distribution of Likelihood Ratio under Nonlinear Hypothesis

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!