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Suppose \(\boldsymbol{Y}\) is an \(n \times 1\) random vector, \(\boldsymbol{X}\) is an \(n \times p\) matrix of known constants of rank \(p\), and \(\beta\) is a \(p \times 1\) vector of regression coefficients. Let \(\boldsymbol{Y}\) have a \(N\left(\boldsymbol{X} \boldsymbol{\beta}, \sigma^{2} \boldsymbol{I}\right)\) distribution. Discuss the joint pdf of \(\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y}\) and \(\boldsymbol{Y}^{\prime}\left[\boldsymbol{I}-\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime}\right] \boldsymbol{Y} / \sigma^{2}\)

Short Answer

Expert verified
The joint pdf of the regression coefficient estimator \(\hat{\boldsymbol{\beta}}\) and the estimator for residual sum of squares \(\boldsymbol{Y}^{\prime}\left[\boldsymbol{I}-\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime}\right] \boldsymbol{Y} / \sigma^{2}\) is the product of their respective distributions. \(\hat{\boldsymbol{\beta}}\) follows a multivariate normal distribution while the residual sum of squares follows a chi-square distribution with n-p degrees of freedom.

Step by step solution

01

Joint Distribution

The distribution of the vector \(\boldsymbol{Y}\) is given by the following equation: \(\boldsymbol{Y} \sim N\left(\boldsymbol{X} \boldsymbol{\beta}, \sigma^{2} \boldsymbol{I}\right)\). The normal equations of the linear regression model are \(\boldsymbol{X}^{\prime}\boldsymbol{X}\hat{\boldsymbol{\beta}} = \boldsymbol{X}^{\prime} \boldsymbol{Y}\), which is the basis of finding \(\hat{\boldsymbol{\beta}}\).
02

Estimator for Regression Coefficients

Solving the normal equations for \(\hat{\boldsymbol{\beta}}\), we find that \(\hat{\boldsymbol{\beta}} = \left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1}\boldsymbol{X}^{\prime} \boldsymbol{Y}\). Therefore, the distribution for \(\hat{\boldsymbol{\beta}}\) is as follows: \(\hat{\boldsymbol{\beta}} \sim N\left(\boldsymbol{\beta}, \sigma^{2}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1}\right)\)
03

Estimator for Residual Sum of Squares

The quantity \(\boldsymbol{Y}^{\prime}\left[\boldsymbol{I}-\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime}\right] \boldsymbol{Y} / \sigma^{2}\) represents the residual sum of squares. It can be shown that, with degrees of freedom n-p, it follows a chi-square distribution, represented as: \(\boldsymbol{Y}^{\prime}\left[\boldsymbol{I}-\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime}\right] \boldsymbol{Y} / \sigma^{2} \sim \chi^{2}_{(n-p)}\)
04

Joint pdf

We need to identify the joint pdf of \(\hat{\boldsymbol{\beta}}\) and \(\boldsymbol{Y}^{\prime}\left[\boldsymbol{I}-\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime}\right] \boldsymbol{Y} / \sigma^{2}\). As these quantities are independent and derived from a normally distributed random vector, the joint pdf would be the product of their individual pdfs, which are normal and chi-square respectively.

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Most popular questions from this chapter

Given the following observations associated with a two-way classification with \(a=3\) and \(b=4\), compute the \(F\) -statistic used to test the equality of the column means \(\left(\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0\right)\) and the equality of the row means \(\left(\alpha_{1}=\alpha_{2}=\alpha_{3}=0\right)\), respectively. $$ \begin{array}{ccccc} \hline \text { Row/Column } & 1 & 2 & 3 & 4 \\ \hline 1 & 3.1 & 4.2 & 2.7 & 4.9 \\ 2 & 2.7 & 2.9 & 1.8 & 3.0 \\ 3 & 4.0 & 4.6 & 3.0 & 3.9 \\ \hline \end{array} $$

Two experiments gave the following results: $$ \begin{array}{cccccc} \hline \mathrm{n} & \bar{x} & \bar{y} & s_{x} & s_{y} & \mathrm{r} \\ \hline 100 & 10 & 20 & 5 & 8 & 0.70 \\ 200 & 12 & 22 & 6 & 10 & 0.80 \\ \hline \end{array} $$ Calculate \(r\) for the combined sample.

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) be \(n\) independent normal variables with common unknown variance \(\sigma^{2}\). Let \(Y_{i}\) have mean \(\beta x_{i}, i=1,2, \ldots, n\), where \(x_{1}, x_{2}, \ldots, x_{n}\) are known but not all the same and \(\beta\) is an unknown constant. Find the likelihood ratio test for \(H_{0}: \beta=0\) against all alternatives. Show that this likelihood ratio test can be based on a statistic that has a well-known distribution.

In Example \(9.1 .2\) verify that \(Q=Q_{3}+Q_{4}\) and that \(Q_{3} / \sigma^{2}\) has a chi-square distribution with \(b(a-1)\) degrees of freedom.

Let \(\mathbf{X}^{\prime}=\left[X_{1}, X_{2}\right]\) be bivariate normal with matrix of means \(\boldsymbol{\mu}^{\prime}=\left[\mu_{1}, \mu_{2}\right]\) and positive definite covariance matrix \(\mathbf{\Sigma}\). Let $$ Q_{1}=\frac{X_{1}^{2}}{\sigma_{1}^{2}\left(1-\rho^{2}\right)}-2 \rho \frac{X_{1} X_{2}}{\sigma_{1} \sigma_{2}\left(1-\rho^{2}\right)}+\frac{X_{2}^{2}}{\sigma_{2}^{2}\left(1-\rho^{2}\right)} $$ Show that \(Q_{1}\) is \(\chi^{2}(r, \theta)\) and find \(r\) and \(\theta\). When and only when does \(Q_{1}\) have a central chi-square distribution?

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