Question

Let \( X_1, X_2, ..., X_n \) be a random sample from \( N(\theta, 1) \), where \( \theta \in (-\infty, \infty) \) is unknown. Consider the problem of testing \( H_0: \theta \le 0 \) against \( H_1: \theta>0 \). Let \( \beta(\theta) \) denote the power function of the likelihood ratio test of size \( \alpha \) (\(0<\alpha<1\)) for testing \( H_0 \) against \( H_1 \). Then, which of the following statements is/are TRUE?

• A. \( \beta(\theta)>\beta(0) \), for all \( \theta>0 \)
• B. \( \beta(\theta)<\beta(0) \), for all \( \theta>0 \)
• C. The critical region of the likelihood test of size \(\alpha\) is \[ \left\{ (x_1, x_2, ..., x_n) \in \mathbb{R}^n : \sqrt{n}\frac{\sum_{i=1}^n x_i}{n}>\tau_{\alpha/2} \right\}, \] where \(\tau_{\alpha/2}\) is a fixed point such that \(P(Z>\tau_{\alpha/2}) = \frac{\alpha}{2}\), \(Z \sim N(0,1)\).
• D. The critical region of the likelihood test of size \(\alpha\) is \[ \left\{ (x_1, x_2, ..., x_n) \in \mathbb{R}^n : \sqrt{n}\frac{\sum_{i=1}^n x_i}{n}>\tau_\alpha \right\}, \] where \(\tau_\alpha\) is a fixed point such that \(P(Z>\tau_\alpha) = \alpha\), \(Z \sim N(0,1)\).

Hint:

For one-sided normal tests, the power function increases with \(\theta\). The critical region is determined by the upper tail of the standard normal distribution.

Answer 1

The Correct Option is A

Step 1: Construct the likelihood ratio test.
Given \( X_i \sim N(\theta, 1) \), the likelihood ratio statistic is \[ \Lambda = \frac{\sup_{\theta \le 0} L(\theta)}{\sup_{\theta} L(\theta)} = \exp\left(-\frac{n}{2}(\bar{X} - \theta)^2 + \frac{n}{2}(\bar{X} - \hat{\theta})^2\right), \] where \( \hat{\theta} = \bar{X} \) (MLE of \( \theta \)). The most powerful test rejects \(H_0\) for large values of \(\bar{X}\). Hence, the critical region is \[ \bar{X}>k, \] for some constant \(k\) determined by the size \(\alpha\).
Step 2: Determine the critical region for size \(\alpha\).
Under \(H_0: \theta = 0\), we have \[ \sqrt{n}(\bar{X} - 0) \sim N(0,1). \] So, \[ P_{H_0}(\bar{X}>k) = P(Z>\sqrt{n}k) = \alpha. \] Therefore, \[ k = \frac{\tau_\alpha}{\sqrt{n}}, \] and the rejection region is \[ \sqrt{n}\bar{X}>\tau_\alpha, \] which matches option (D).
Step 3: Analyze the power function.
For \(\theta>0\), the test statistic shifts rightward, so \[ \beta(\theta) = P_\theta(\bar{X}>k)>P_0(\bar{X}>k) = \beta(0), \] hence (A) is true. All other options are incorrect or misstate the critical value. Final Answer: \[ \boxed{(A) \text{ and } (D)} \]