Consider $X_{1}, \ldots, X_{n}$ from a $N\left(\mu, \sigma^{2}\right)$ distribution with parameter $\theta=\left(\mu, \sigma^{2}\right) \in$ $\Theta=\mathbb{R} \times(0, \infty)$. Derive the likelihood ratio test statistic $\Lambda_{n}\left(\Theta, \Theta_{0}\right)$ for the composite hypothesis

$$H_{0}: \sigma^{2}=1 \text { vs. } H_{1}: \sigma^{2} \neq 1$$

where $\Theta_{0}=\{(\mu, 1): \mu \in \mathbb{R}\}$ is the parameter space constrained by $H_{0}$.

Prove carefully that

$$\Lambda_{n}\left(\Theta, \Theta_{0}\right) \rightarrow^{d} \chi_{1}^{2} \quad \text { as } n \rightarrow \infty$$

where $\chi_{1}^{2}$ is a Chi-Square distribution with one degree of freedom.