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Chapter 9: Problem 4

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

Short Answer

Expert verified
The square of a noncentral \(T\) random variable is equivalent to a noncentral \(F\) random variable.

Step by step solution

01

Definition of Noncentral \(T\) Distribution

Let's define a noncentral \(T\) random variable. It can be expressed as \(T = \frac{N(0,1) + \lambda}{\sqrt{X/1}}\) where \(N(0,1)\) is a standard normal random variable, \(X\) follows a chi-square distribution with degree of freedom 1, and \(\lambda\) is the noncentrality parameter.
02

Squaring the Noncentral \(T\) variable

Squaring the \(T\) variable yields \(T^2 = (\frac{N(0,1) + \lambda}{\sqrt{X/1}})^2\) which simplifies to \(T^2 = \frac{{(N(0,1) + \lambda)}^2}{X}\). In other words, the square of a noncentral \(T\) variable is a ratio of a squared normal variable shifted by \(\lambda\) to a chi-square distributed variable.
03

Recognizing the Noncentral \(F\) Distribution

Upon inspection, we can see that the ratio matches the ratio for a noncentral \(F\) distribution. A noncentral \(F\) distribution with 1 and \(k\) degrees of freedom can be defined as \(F = \frac{{Y+\lambda}^2}{X/k}\). When \(k=1\), \(F = \frac{{Y+\lambda}^2}{X}\), which is equivalent to \(T^2\), and thus the square of noncentral \(T\) random variable is a noncentral \(F\) random variable.

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Most popular questions from this chapter

Let \(A\) be the real symmetric matrix of a quadratic form \(Q\) in the observations of a random sample of size \(n\) from a distribution which is \(N\left(0, \sigma^{2}\right)\). Given that \(Q\) and the mean \(\bar{X}\) of the sample are independent, what can be said of the elements of each row (column) of \(\boldsymbol{A}\) ? Hint: Are \(Q\) and \(\bar{X}^{2}\) independent?

Let \(X_{1}, X_{2}, X_{3}, X_{4}\) denote a random sample of size 4 from a distribution which is \(N\left(0, \sigma^{2}\right) .\) Let \(Y=\sum_{1}^{4} a_{i} X_{i}\), where \(a_{1}, a_{2}, a_{3}\), and \(a_{4}\) are real constants. If \(Y^{2}\) and \(Q=X_{1} X_{2}-X_{3} X_{4}\) are independent, determine \(a_{1}, a_{2}, a_{3}\), and \(a_{4} .\)

Suppose \(\mathbf{X}\) is an \(n \times p\) matrix with rank \(p\). (a) Show that \(\operatorname{ker}\left(\mathbf{X}^{\prime} \mathbf{X}\right)=\operatorname{ker}(\mathbf{X})\). (b) Use part (a) and the last exercise to show that if \(\mathbf{X}\) has full column rank, then \(\mathbf{X}^{\prime} \mathbf{X}\) is nonsingular.

Show that $$ \sum_{j=1}^{b} \sum_{i=1}^{a}\left(X_{i j}-\bar{X}_{i .}\right)^{2}=\sum_{j=1}^{b} \sum_{i=1}^{a}\left(X_{i j}-\bar{X}_{i .}-\bar{X}_{. j}+\bar{X}_{. .}\right)^{2}+a \sum_{j=1}^{b}\left(\bar{X}_{. j}-\bar{X}_{. .}\right)^{2} . $$

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 \(\boldsymbol{\Sigma}\). 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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