題組內容

2. (50%) Let $X1, X2, \cdots, X_n$ be a random sample from a normal distribution

$$f(x; \theta) = (2\pi\theta)^{-1/2} \exp \left( -\frac{1}{2\theta} (x - \mu)^2 \right), x \in R; \mu \in R, \theta > 0.$$

Let the distribution function of $f(x; \theta)$ be denoted as $\Phi(x; \mu, \theta)$.

(e) (10%) Find the Likelihood ratio test (LR test) for testing \( H0: \theta = \theta0 \) versus \( H0: \theta \neq \theta0 \), where \( \theta_0 \) is a pre-specified value and provide the corresponding critical value for the LR test.