Question

Let the independent random variables \(Y_{1}, \ldots, Y_{n}\) have the joint \(\mathrm{pdf}\) $$ L\left(\alpha, \beta, \sigma^{2}\right)=\left(\frac{1}{2 \pi \sigma^{2}}\right)^{n / 2} \exp \left\\{-\frac{1}{2 \sigma^{2}} \sum_{1}^{n}\left[y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}\right\\} $$ where the given numbers \(x_{1}, x_{2}, \ldots, x_{n}\) are not all equal. Let \(H_{0}: \beta=0(\alpha\) and \(\sigma^{2}\) unspecified). It is desired to use a likelihood ratio test to test \(H_{0}\) against all possible alternatives. Find \(\Lambda\) and see whether the test can be based on a familiar statistic. Hint: In the notation of this section, show that $$ \sum_{1}^{n}\left(Y_{i}-\hat{\alpha}\right)^{2}=Q_{3}+\widehat{\beta}^{2} \sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} $$

Short answer

Finding the likelihood ratio test statistic involves analyzing the given formula, calculating the statistic \(\Lambda\), simplifying the sum in the likelihood function, and understanding if the test can be based on a familiar statistic. The full calculation of \(\Lambda\) and its relationship to known statistics depends on the specific values of the observations and parameters.

Step-by-step solution

Analyze Given Formula

The provided formula represents the likelihood function of a linear regression model with \(n\) observations, with \(\alpha\) as the intercept, \(\beta\) as the slope, \(\sigma^2\) as the variance of errors and \(y_i\) and \(x_i\) being the dependent and independent variables, respectively. The bar notation \(\overline{x}\) denotes the mean of \(x\). The sum ranges over all observations \(i=1, \ldots, n\). The task is to evaluate the likelihood function for \(\beta = 0\).

Calculation of \(\Lambda\)

The likelihood ratio statistic \(\Lambda\) is calculated as the ratio of the likelihood under the null hypothesis (\(H_0: \beta=0\)) to the likelihood under the alternative (\(H_1: \beta \neq 0\)). Given the likelihood function \(L(\alpha, \beta, \sigma^{2})\), \(\Lambda\) can be calculated as \(L(\alpha, 0, \sigma_{0}^{2})/L(\alpha, \hat{\beta}, \hat{\sigma}^{2})\). Here, \((\alpha, 0, \sigma_{0}^{2})\) are the parameter values under \(H_0\) and \((\alpha, \hat{\beta}, \hat{\sigma}^{2})\) are those that maximize the likelihood under \(H_1\)

Simplify Summation

To simplify the sum \(\sum_{1}^{n}\left(y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right)^{2}\), use the hint that shows it can be rewritten as \(\sum_{1}^{n}\left(Y_{i}-\hat{\alpha}\right)^{2}=Q_{3}+\widehat{\beta}^{2}\sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2}\). Here, \(\hat{\alpha}\) and \(\hat{\beta}\) are the estimated values of \(\alpha\) and \(\beta\) that maximize the likelihood under \(H_1\)

Determine Whether Test Can be Based on a Familiar Statistic

After calculating \(\Lambda\) and simplifying, examine the likelihood ratio statistic for its relationship to known statistics. It may be related to a Chi-square, F, or T statistic, for example.