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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

Expert verified
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

01

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.
02

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.
03

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 \(\Lambda\).
04

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.

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Most popular questions from this chapter

Using the background of the two-way classification with one observation per cell, show that the maximum likelihood estimator of \(\alpha_{i}, \beta_{j}\), and \(\mu\) are \(\hat{\alpha}_{i}=\bar{X}_{i .}-\bar{X}_{. .}\) \(\hat{\beta}_{j}=\bar{X}_{. j}-\bar{X}_{. .}\), and \(\hat{\mu}=\bar{X}_{. .}\), respectively. Show that these are unbiased estimators of their respective parameters and compute \(\operatorname{var}\left(\hat{\alpha}_{i}\right), \operatorname{var}\left(\hat{\beta}_{j}\right)\), and \(\operatorname{var}(\hat{\mu})\).

Let \(X_{1}, X_{2}, X_{3}\) be a random sample from the normal distribution \(N\left(0, \sigma^{2}\right)\). Are the quadratic forms \(X_{1}^{2}+3 X_{1} X_{2}+X_{2}^{2}+X_{1} X_{3}+X_{3}^{2}\) and \(X_{1}^{2}-2 X_{1} X_{2}+\frac{2}{3} X_{2}^{2}-\) \(2 X_{1} X_{2}-X_{3}^{2}\) independent or dependent?

Student's scores on the mathematics portion of the ACT examination, \(x\), and on the final examination in the first-semester calculus ( 200 points possible), \(y\), are given. (a) Calculate the least squares regression line for these data. (b) Plot the points and the least squares regression line on the same graph. (c) Find point estimates for \(\alpha, \beta\), and \(\sigma^{2}\). (d) Find 95 percent confidence intervals for \(\alpha\) and \(\beta\) under the usual assumptions. $$ \begin{array}{cc|cc} \hline \mathrm{x} & \mathrm{y} & \mathrm{x} & \mathrm{y} \\ \hline 25 & 138 & 20 & 100 \\ 20 & 84 & 25 & 143 \\ 26 & 104 & 26 & 141 \\ 26 & 112 & 28 & 161 \\ 28 & 88 & 25 & 124 \\ 28 & 132 & 31 & 118 \\ 29 & 90 & 30 & 168 \\ 32 & 183 & & \\ \hline \end{array} $$

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a normal distribution \(N\left(\mu, \sigma^{2}\right)\). Show that $$ \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}=\sum_{i=2}^{n}\left(X_{i}-\bar{X}^{\prime}\right)^{2}+\frac{n-1}{n}\left(X_{1}-\bar{X}^{\prime}\right)^{2}, $$ where \(\bar{X}=\sum_{i=1}^{n} X_{i} / n\) and \(\bar{X}^{\prime}=\sum_{i=2}^{n} X_{i} /(n-1)\). Hint: \(\quad\) Replace \(X_{i}-\bar{X}\) by \(\left(X_{i}-\bar{X}^{\prime}\right)-\left(X_{1}-\bar{X}^{\prime}\right) / n\). Show that \(\sum_{i=2}^{n}\left(X_{i}-\bar{X}^{\prime}\right)^{2} / \sigma^{2}\) has a chi-square distribution with \(n-2\) degrees of freedom. Prove that the two terms in the right-hand member are independent. What then is the distribution of $$ \frac{[(n-1) / n]\left(X_{1}-\bar{X}^{\prime}\right)^{2}}{\sigma^{2}} ? $$

Show that the square of a noncentral \(T\) random variable is a noncentral \(F\) random variable.

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