Likelihood Ratio Test and Wilks' Asymptotic Theorem

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Asymptotic Hypothesis Testing.

Your Task
Let $X_1, \dots, X_n \sim f(x \mid \theta)$ where $\theta \in \Theta \subseteq \mathbb{R}^p$. We test the composite null hypothesis $H_0: \theta \in \Theta_0$ against $H_1: \theta \in \Theta \setminus \Theta_0$, where $\Theta_0$ is a subspace of dimension $p - r$ (specifying $r$ constraints).
(1) Define the likelihood ratio test statistic $\Lambda = \frac{\sup_{\theta \in \Theta_0} L(\theta)}{\sup_{\theta \in \Theta} L(\theta)}$ and deviance $D = -2 \ln \Lambda$.
(2) Prove Wilks' Theorem: under $H_0$ and standard Cramér regularity conditions, $D \xrightarrow{d} \chi^2(r)$ as $n \to \infty$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.