Chapter 9: Problem 4
Show that the square of a noncentral \(T\) random variable is a noncentral \(F\) random variable.
Chapter 9: Problem 4
Show that the square of a noncentral \(T\) random variable is a noncentral \(F\) random variable.
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Get started for freeUsing the notation of Section 9.2, assume that the means \(\mu_{j}\) satisfy a linear function of \(j\), nanely \(\mu_{j}=c+d[j-(b+1) / 2] .\) Let independent random samples of size \(a\) be taken from the \(b\) normal distributions having means \(\mu_{1}, \mu_{2}, \ldots, \mu_{b}\), respectively, and common unknown variance \(\sigma^{2}\). (a) Show that the maximum likelihood estimators of \(c\) and \(d\) are, respectively, \(\hat{c}=\bar{X}_{. .}\) and $$ \hat{d}=\frac{\sum_{j=1}^{b}[j-(b-1) / 2]\left(\bar{X}_{. j}-\bar{X}_{. .}\right)}{\sum_{j=1}^{b}[j-(b+1) / 2]^{2}} $$ (b) Show that $$ \begin{aligned} \sum_{i=1}^{a} \sum_{j=1}^{b}\left(X_{i j}-\bar{X}_{. .}\right)^{2}=\sum_{i=1}^{a} \sum_{j=1}^{b}\left[X_{i j}-\bar{X}_{. .}-\hat{d}\left(j-\frac{b+1}{2}\right)\right]^{2} \\ &+\hat{d}^{2} \sum_{j=1}^{b} a\left(j-\frac{b+1}{2}\right)^{2} \end{aligned} $$ (c) Argue that the two terms in the right-hand member of Part (b), once divided by \(\sigma^{2}\), are independent random variables with \(\chi^{2}\) distributions provided that \(d=0\) (d) What \(F\) -statistic would be used to test the equality of the means, that is, \(H_{0}: d=0 ?\)
If \(A_{1}, A_{2}, \ldots, A_{k}\) are events, prove, by induction, Boole's inequality $$ P\left(A_{1} \cup A_{2} \cup \cdots \cup A_{k}\right) \leq \sum_{1}^{k} P\left(A_{i}\right) $$ Then show that $$ P\left(A_{1}^{c} \cap A_{2}^{c} \cap \cdots \cap A_{k}^{c}\right) \geq 1-\sum_{1}^{b} P\left(A_{i}\right) $$
Suppose \(\mathbf{A}\) is a real symmetric matrix. If the eigenvalues of \(\mathbf{A}\) are only 0 's and 1 's then prove that \(\mathbf{A}\) is idempotent.
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 the \(4 \times 1\) matrix \(\boldsymbol{Y}\) be multivariate normal \(N\left(\boldsymbol{X} \boldsymbol{\beta}, \sigma^{2} \boldsymbol{I}\right)\), where the \(4 \times 3\) matrix \(\boldsymbol{X}\) equals $$ \boldsymbol{X}=\left[\begin{array}{rrr} 1 & 1 & 2 \\ 1 & -1 & 2 \\ 1 & 0 & -3 \\ 1 & 0 & -1 \end{array}\right] $$ and \(\beta\) is the \(3 \times 1\) regression coeffient matrix. (a) Find the mean matrix and the covariance matrix of \(\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y}\). (b) If we observe \(\boldsymbol{Y}^{\prime}\) to be equal to \((6,1,11,3)\), compute \(\hat{\boldsymbol{\beta}}\).
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