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Chapter 9: Problem 2

Under Model \((9.2 .1)\), show that the linear functions \(X_{i j}-\bar{X}_{. j}\) and \(\bar{X}_{. j}-\bar{X}_{\cdots}\) are uncorrelated. Hint: Recall the definition of \(\bar{X}_{. j}\) and \(\bar{X}_{. .}\) and, without loss of generality, we can let \(E\left(X_{i j}\right)=0\) for all \(i, j\).

Short Answer

Expert verified
The linear functions \(X_{i j}-\bar{X}_{. j}\) and \(\bar{X}_{. j}-\bar{X}_{..}\) are uncorrelated because their covariance is zero.

Step by step solution

01

Recall Definitions

Recall that \(\bar{X}_{. j}=\frac{1}{n} \sum_{i=1}^{n} X_{i j}\) is the sample mean of the \(j^{t h}\) column, and \(\bar{X}_{..}=\frac{1}{n p} \sum_{i=1}^{n} \sum_{j=1}^{p} X_{i j}\) is the grand mean.
02

Compute Covariance

Compute the covariance of \(X_{i j}-\bar{X}_{. j}\) and \(\bar{X}_{. j}-\bar{X}_{..}\). Use the formula for covariance: \[Cov(X, Y) = E[(X - E[X])(Y - E[Y])]\] Remember that we're given that \(E\left(X_{i j}\right)=0\) for all \(i, j\).
03

Simplify the Covariance

Simplify the covariance expression obtained in step 2. Apply the linearity of expectation and break it down further.
04

Conclude Uncorrelation

Conclude that the two linear functions are uncorrelated if their covariance is equal to zero. This is done by showing that the expression in step 3 simplifies to zero.

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Most popular questions from this chapter

Let \(A_{1}, A_{2}, \ldots, A_{k}\) be the matrices of \(k>2\) quadratic forms \(Q_{1}, Q_{2}, \ldots, Q_{k}\) in the observations of a random sample of size \(n\) from a distribution that is \(N\left(0, \sigma^{2}\right)\). Prove that the pairwise independence of these forms implies that they are mutually independent. Hint: \(\quad\) Show that \(\boldsymbol{A}_{i} \boldsymbol{A}_{j}=\mathbf{0}, i \neq j\), permits \(E\left[\exp \left(t_{1} Q_{1}+t_{2} Q_{2}+\cdots+t_{k} Q_{k}\right)\right]\) to be written as a product of the mgfs of \(Q_{1}, Q_{2}, \ldots, Q_{k}\).

For the two-way interaction model, \((9.5 .15)\), show that the following decomposition of sums of squares is true: $$ \begin{aligned} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{\ldots}\right)^{2}=& b c \sum_{i=1}^{a}\left(\bar{X}_{i . .}-\bar{X}_{\ldots .}\right)^{2}+a c \sum_{j=1}^{b}\left(\bar{X}_{. j .}-\bar{X}_{\ldots}\right)^{2} \\ &+c \sum_{i=1}^{a} \sum_{j=1}^{b}\left(\bar{X}_{i j .}-\bar{X}_{i . .}-\bar{X}_{. j .}+\bar{X}_{\ldots}\right)^{2} \\ &+\sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{i j .}\right)^{2} \end{aligned} $$ that is, the total sum of squares is decomposed into that due to row differences, that due to column differences, that due to interaction, and that within cells.

Students' 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: $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 25 & 20 & 26 & 26 & 28 & 28 & 29 & 32 & 20 & 25 \\ \hline y & 138 & 84 & 104 & 112 & 88 & 132 & 90 & 183 & 100 & 143 \\ \hline x & 26 & 28 & 25 & 31 & 30 & & & & & \\ \hline y & 141 & 161 & 124 & 118 & 168 & & & & & \\ \hline \end{array} $$ The data are also in the rda file regr1.rda. Use \(\mathrm{R}\) or another statistical package for computation and plotting. (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) Obtain the residual plot and comment on the appropriateness of the model. (d) Find \(95 \%\) confidence interval for \(\beta\) under the usual assumptions. Comment in terms of the problem.

Let \(\boldsymbol{X}^{\prime}=\left[X_{1}, X_{2}, \ldots, X_{n}\right]\), where \(X_{1}, X_{2}, \ldots, X_{n}\) are observations of a random sample from a distribution that is \(N\left(0, \sigma^{2}\right) .\) Let \(b^{\prime}=\left[b_{1}, b_{2}, \ldots, b_{n}\right]\) be a real nonzero vector, and let \(\boldsymbol{A}\) be a real symmetric matrix of order \(n\). Prove that the linear form \(b^{\prime} X\) and the quadratic form \(\boldsymbol{X}^{\prime} \boldsymbol{A} \boldsymbol{X}\) are independent if and only if \(\boldsymbol{b}^{\prime} \boldsymbol{A}=\mathbf{0}\). Use this fact to prove that \(\boldsymbol{b}^{\prime} \boldsymbol{X}\) and \(\boldsymbol{X}^{\prime} \boldsymbol{A} \boldsymbol{X}\) are independent if and only if the two quadratic forms \(\left(\boldsymbol{b}^{\prime} \boldsymbol{X}\right)^{2}=\boldsymbol{X}^{\prime} \boldsymbol{b b}^{\prime} \boldsymbol{X}\) and \(\boldsymbol{X}^{\prime} \boldsymbol{A} \boldsymbol{X}\) are independent.

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