Question

Let \( X_1, X_2, \dots, X_n \) be a random sample from a \( N(\theta, 1) \) distribution. Instead of observing \( X_1, X_2, \dots, X_n \), we observe \( Y_i = e^{X_i}, i = 1, 2, \dots, n \). To test the hypothesis \[ H_0: \theta = 1 \quad \text{against} \quad H_1: \theta \neq 1 \] based on the random sample \( Y_1, Y_2, \dots, Y_n \), the rejection region of the likelihood ratio test is of the form, for some \( c_1<c_2 \),

• A. \( \sum_{i=1}^{n} Y_i \leq c_1 \) or \( \sum_{i=1}^{n} Y_i \geq c_2 \)
• B. \( c_1 \leq \sum_{i=1}^{n} Y_i \leq c_2 \)
• C. \( c_1 \leq \sum_{i=1}^{n} \log Y_i \leq c_2 \)
• D. \( \sum_{i=1}^{n} \log Y_i \leq c_1 \) or \( \sum_{i=1}^{n} \log Y_i \geq c_2 \)

Hint:

For hypothesis testing involving transformations of data, the likelihood ratio test often involves sums of transformed variables. In this case, the rejection region is determined by the sum of the logs of the transformed variables.

Answer 1

The Correct Option is D

Step 1: Likelihood ratio test setup.
In a likelihood ratio test, we compare the likelihood of the data under the null hypothesis \( H_0 \) and the alternative hypothesis \( H_1 \). The likelihood ratio test statistic is given by the ratio of the likelihood under \( H_1 \) to the likelihood under \( H_0 \). Step 2: The transformation of the data.
We are given the transformation \( Y_i = e^{X_i} \). The likelihood function for \( Y_1, Y_2, \dots, Y_n \) is based on the transformed data. The test statistic involves the sum of the logs of \( Y_i \), i.e., \( \sum_{i=1}^{n} \log Y_i \). Step 3: Deriving the rejection region.
Under \( H_0 \), the distribution of the test statistic \( \sum_{i=1}^{n} \log Y_i \) follows a certain distribution. The rejection region for the test is determined by the values of \( \sum_{i=1}^{n} \log Y_i \) falling outside the acceptance region, which is of the form: \[ \sum_{i=1}^{n} \log Y_i \leq c_1 \quad \text{or} \quad \sum_{i=1}^{n} \log Y_i \geq c_2 \] for some constants \( c_1 \) and \( c_2 \). Step 4: Conclusion.
The correct answer is (D) \( \sum_{i=1}^{n} \log Y_i \leq c_1 \) or \( \sum_{i=1}^{n} \log Y_i \geq c_2 \).