This is from George Casella textbook question 8.37 (c). Let $X_1,…,X_n$ be a random sample from a $n(\theta, \sigma^2)$ population. Consider testing $H_0:\theta\leq \theta_0$ versus $H_1:\theta> \theta_0$. If $\sigma^2$ is unknown, what is the LRT statistic? The solution is $\lambda=(\hat{\sigma}^2/\hat{\sigma_0}^2)^{n/2}$. I didn't get this.
My attempt: the LRT of denimoator is easier and reached when $\theta=\bar{x}$ and $\sigma^2=\hat{\sigma}^2$ below in the solution. The numerator is $\sigma^2=\hat{\sigma}^2$ and $\theta=\theta_0$.(We don't need to consider when $\lambda=1$ case.) I didn't get the LRT statistic in the solution.
Best Answer
According to Casella & Berger's notation (Definition 8.2.1), the likelihood ratio test statistic for testing $H_0: \theta \in \Theta_0$ versus $H_1: \theta \in \Theta_0^c$ is $$\lambda(\boldsymbol{x}) = \frac{\text{sup}_{\Theta_0}L(\theta \mid \boldsymbol{x})}{\text{sup}_{\Theta}L(\theta \mid \boldsymbol{x})}.$$ A likelihood ratio test (LRT) is any test that has a rejection region of the form $\{\boldsymbol{x}: \lambda(\boldsymbol{x}) \leq c\}$, where $c$ is any number satisfying $0 \leq c \leq 1$.
In the problem of testing $H_0: \theta \leq \theta_0$ versus $H_1: \theta > \theta_0$, where $X_1, \dots, X_n \overset{i.i.d.}{\sim} n(\theta, \sigma^2)$ and $\sigma^2$ is unknown, we have $\Theta_0 = \{(\theta, \sigma^2): \theta \leq \theta_0, \sigma^2 > 0\}$ and $\Theta = \{(\theta, \sigma^2): \theta \in \mathbb{R}, \sigma^2 > 0\}$, and the likelihood is given by $$L(\theta, \sigma^2 \mid \boldsymbol{x}) = \left(\frac{1}{\sqrt{2\pi\sigma^2}}\right)^n \exp\left(-\frac{1}{2\sigma^2}\sum_{i=1}^n(x_i - \theta)^2\right).$$
The supremum in the denominator of $\lambda(\boldsymbol{x})$ is attained at the unrestricted MLE of $(\theta, \sigma^2)$, that is, $\hat{\theta} = \bar{x}$ and $\hat{\sigma}^2 = \frac{1}{n}\sum_{i=1}^n(x_i - \bar{x})^2$ (See Section 4.1 if this is unclear).
The supremum in the numerator of $\lambda(\boldsymbol{x})$ needs to be a bit more careful, since it is "restricted" to the parameter space $\Theta_0$. Specifically, if we observe $\bar{x} > \theta_0$ ("$>$" is more precise than "$\geq$" the solution), then $L(\theta, \sigma^2)$ cannot be maximized at the unrestricted MLE ($\hat{\theta} = \bar{x}, \hat{\sigma}^2 = \frac{1}{n}\sum_{i=1}^n(x_i - \bar{x})^2$) since $(\bar{x}, \hat{\sigma}^2) \notin \Theta_0$. In this case, the restricted MLE (with respect to $\Theta_0$) of $(\theta, \sigma^2)$ is taken at the boundary of the parameter space, that is, $\hat{\theta}_0 = \theta_0$ and $\hat{\sigma}_0^2 = \frac{1}{n}\sum_{i=1}^n(x_i - \theta_0)^2$. On the other hand, if we observe $\bar{x} \leq \theta_0$, then $L(\theta, \sigma^2)$ can be maximized at the unrestricted MLE ($\hat{\theta} = \bar{x}, \hat{\sigma}^2 = \frac{1}{n}\sum_{i=1}^n(x_i - \bar{x})^2$) since $(\bar{x}, \hat{\sigma}^2) \in \Theta_0$. In this case, the numerator and denominator coincides thus $\lambda(\boldsymbol{x}) = 1$.
To sum up, for $n(\theta, \sigma^2)$ where $\sigma^2$ is unknown, the LRT statistic of testing $H_0: \theta \leq \theta_0$ versus $H_1: \theta > \theta_0$ is given by $$\lambda(\boldsymbol{x}) = \begin{cases} \dfrac{\text{sup}_{\Theta_0}L(\theta, \sigma^2 \mid \boldsymbol{x})}{\text{sup}_{\Theta}L(\theta, \sigma^2 \mid \boldsymbol{x})} = \dfrac{L(\theta_0, \hat{\sigma}_0^2 \mid \boldsymbol{x})}{L(\bar{x}, \hat{\sigma}^2 \mid \boldsymbol{x})} = \left(\dfrac{\hat{\sigma}^2}{\hat{\sigma}_0^2}\right)^{n/2} & \text{if } \bar{x} > \theta_0; \\ 1 & \text{if } \bar{x} \leq \theta_0. \end{cases}$$
Remark: It is worth to note the fact that $$\exp\left(-\frac{1}{2\hat{\sigma}_0^2}\sum_{i=1}^n(x_i - \theta_0)^2\right) = \exp\left(-\frac{1}{2\hat{\sigma}^2}\sum_{i=1}^n(x_i - \bar{x})^2\right) = \exp\left(-\frac{n}{2}\right),$$ which cancels in the derivation of the LRT statistic $\lambda(\boldsymbol{x})$ for $\bar{x} > \theta_0$.