Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be iid \(N\left(\theta_{1}, \theta_{2}\right) .\) Show that the likelihood ratio principle for testing \(H_{0}: \theta_{2}=\theta_{2}^{\prime}\) specified, and \(\theta_{1}\) unspecified, against \(H_{1}: \theta_{2} \neq \theta_{2}^{\prime}, \theta_{1}\) unspecified, leads to a test that rejects when \(\sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} \leq c_{1}\) or \(\sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} \geq c_{2}\) where \(c_{1}

Short answer

The problem solves the hypothesis testing problem using the likelihood ratio principle. The likelihood ratio test principle rejects the null hypothesis \( H_{0} \) when the sum of squared deviations, which is the maximum likelihood estimator of the variance under \( H_{0} \), is either smaller or larger than two selected constants \( c_{1} \) and \( c_{2} \) respectively, where \( c_{1} < c_{2} \).

Step-by-step solution

Understanding the likelihood ratio test principle

The likelihood ratio for a test of hypothesis \( H_{0} \) versus \( H_{1} \) is defined as: \( \Lambda = \frac{L(\theta_{2}^{\prime})}{L(\theta_{2})} \), where \( L(\theta) \) is the likelihood function which measures how likely the observed data are for different values of \( \theta \), and \( \theta \) and \( \theta' \) represent the parameter under \( H_{0} \) and \( H_{1} \) respectively.

Define the test statistic

Under the null hypothesis \( H_{0} \), the maximum likelihood estimator of the variance \( \sigma^2 \) of a normal distribution is the sample variance \( S^2 \), defined as \( S^2 = \sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} \). This will be our test statistic.

Determine the rejection regions

The Likelihood Ratio Test (LRT) tends to be most powerful when the likelihood under \( H_{1} \) is furthest from \( H_{0} \). Because of this, the LRT will tend to reject \( H_{0} \) when our test statistic is significantly greater or significantly less than what we would expect if the null hypothesis were true. This means that our rejection region will be when \( S^2 \leq c_{1} \) or \( S^2 \geq c_{2} \) where \( c_{1} < c_{2} \) are selected appropriately depending on the significance level of the test.