In the book "All of statistics" (Wasserman 2005) they present the likelihood ratio hypothesis test, using the likelihood ratio statistic: $$\lambda=2\log{\Lambda\left(x^{n}\right)\triangleq2\log {\frac{\sup_{\theta\in\Theta}L\left(\theta\right)}{\sup_{\theta\in\Theta_{0}}L\left(\theta\right)}}}=2\log {\frac{\mathcal{L}(\hat\theta)}{\mathcal{L}(\hat\theta_0)}}$$ They also present the Wilks theorem stating that under $H_0:\theta\in\Theta_0$: $$\lambda\left(X^{n}\right)\triangleq2\ln\Lambda\left(X^{n}\right)\rightsquigarrow\chi_{d}^{2}$$ where $d$ is the dimension difference of $\Theta$ and $\Theta_0$.

However, it seems extremely odd that a Chi-square RV can take negative values (when the likelihood for $H_0$ is larger than the likelihood of $H_1$).


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Checked the original paper "The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses" (Wilks 1938).

Turns out they limit this theorem to cases where $\Theta_0$ is a subset of $\Theta$, implying $\frac{\sup_{\theta\in\Theta}L\left(\theta\right)}{\sup_{\theta\in\Theta_{0}}L\left(\theta\right)}\ge 1$ and thus $\lambda(X^n)\ge 0$.


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