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.

Key concepts

Independent Random Variables

When dealing with independent random variables, it's important to understand that each variable's outcome does not affect the others. In our exercise, we consider the variables \(Y_1, Y_2, \ldots, Y_n\), assuming they are independent. This means:

• The probability distribution of any one of the \(Y_i\) is unaffected by the values of the others.
• Calculations for probabilities can be simplified, as we can handle each variable separately.

Understanding independence allows us to construct joint probability density functions more easily. This is crucial because it simplifies the likelihood functions we use in hypothesis testing, including the likelihood ratio test as depicted in the step-by-step solution.

Joint Probability Density Function

The joint probability density function (pdf) is a function used to describe the probability distribution of multiple random variables. In the context of this exercise, we're dealing with the joint pdf of the independent random variables \(Y_1, \ldots, Y_n\). The joint pdf is given by:\[L\left(\alpha, \beta, \sigma^{2}\right)=\left(\frac{1}{2 \pi \sigma^{2}}\right)^{n / 2} \exp \left\{-\frac{1}{2 \sigma^{2}} \sum_{1}^{n}\left[y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}\right\}\]Key points about joint pdf:

• It describes the likelihood of observing a particular set of values for all variables simultaneously.
• For independent variables, the joint pdf simplifies to the product of the individual pdfs.
• It is foundational in evaluating likelihoods under different hypotheses in statistical inference.

The joint pdf encapsulates all the needed information to assess both the null and alternative hypotheses effectively by calculating the likelihoods corresponding to each scenario.

Null and Alternative Hypotheses

In hypothesis testing, we often set up a null hypothesis \(H_0\) and an alternative hypothesis \(H_1\). These are statements about the probability distributions of the random variables under different assumptions. For this exercise:

• The null hypothesis \(H_0\) assumes that the parameter \(\beta\) is zero, which simplifies the model to not include the effect of \(\beta\).
• The alternative hypothesis considers \(\beta\) not equal to zero, which suggests that \(\beta\) may have an effect on the response \(Y\).

The essence of hypothesis testing is assessing whether the collected data provides sufficient evidence to reject the null hypothesis in favor of the alternative. Here, the likelihood ratio \(\Lambda\) helps decide this by comparing the likelihoods of data under both hypotheses.

Statistical Inference

Statistical inference involves drawing conclusions about population parameters based on sample data. In this context, we're employing the likelihood ratio test, a method of statistical inference. This test helps determine whether the null hypothesis can be rejected.The likelihood ratio test:

• Calculates the ratio of likelihood functions under the two hypotheses (null and alternative).
• Uses this ratio to infer which hypothesis is more likely to be true given the data.
• Often compares \(\Lambda\) to a threshold derived from statistical tables (e.g., chi-square distribution), to decide whether to reject the null hypothesis.

By understanding statistical inference, one appreciates how data-driven decisions are made, allowing for less intuitive but mathematically sound conclusions about the populations being studied.