Question

Let the independent random variables \(Y_{1}, \ldots, Y_{n}\) have the joint 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

The exercise is mainly about applying the likelihood ratio test to test the null hypothesis against all possible alternatives and identify a well-known statistic that the test might be based on. In-depth understanding of likelihood ratio tests as well as the significance of the given hint will greatly facilitate the successful completion of the exercise.

Step-by-step solution

Write out the Likelihood Functions under Null and Alternative Hypotheses

First, formulate the likelihood function under the null hypothesis \(H_{0}\) (\(\beta=0\)) and the alternative hypothesis (\(\beta\neq0\)). The likelihood function under the null hypothesis is denoted by: \[L_{0}\left(\alpha, \sigma^{2}\right)=\left(\frac{1}{2 \pi \sigma^{2}}\right)^{n / 2} \exp \left\{-\frac{1}{2 \sigma^{2}} \sum_{1}^{n}\left[y_{i}-\alpha\right]^{2}\right\}\], while the likelihood function under the alternative hypothesis is the one provided in the problem.

Compute the Likelihood Ratio

Compute the likelihood ratio, which is defined as the ratio of the likelihoods under the null and the alternative hypotheses. \[\Lambda=\frac{L_{0}\left(\alpha, \sigma^{2}\right)}{L\left(\alpha, \beta, \sigma^{2}\right)}\] This forms the basis for the likelihood ratio test.

Apply the Given Hint

Make good use of the given hint which states \[\sum_{1}^{n}\left(Y_{i}-\hat{\alpha}\right)^{2}=Q_{3}+\widehat{\beta}^{2} \sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2}\] This hint helps to further simplify the expression for the likelihood ratio \(\Lambda\).

Identify Familiar Statistic

After simplifying the expression for \(\Lambda\), identify a commonly known statistic (like the \(t\)-statistic, \(F\)-statistic, etc.) by comparing it with standard formulas.