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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 a noncentral \(F\) random variable. This is due to the underlying mathematical definitions and properties of these distributions.

Step by step solution

01

Define a Noncentral \(T\) Distribution

A noncentral \(T\) variable may be defined as the ratio \(T = X / \sqrt{Y}\), where \(X\) follows a normal distribution with mean \(\mu\) and standard deviation \(1\), and \(Y\) follows a chi-square distribution with one degree of freedom. It's essential to note that \(X\) and \(Y\) are independent random variables.
02

Square the Noncentral \(T\) Distribution

Squaring the \(T\) random variable yields \(T^2 = X^2 / Y\). The \(X^2\) term is a chi-square distribution with one degree of freedom, and \(Y\) is a chi-square distribution with one degree of freedom, based on the definition from Step 1.
03

Define a Noncentral \(F\) Distribution

A noncentral \(F\) distribution is defined as the ratio of two independent chi-square random variables each divided by their respective degrees of freedom. In other words, let \(Z_1\) and \(Z_2\) be independent random variables each following chi-square distribution with degrees of freedom \(d_1\) and \(d_2\) respectively, then the noncentral \(F\) distribution can be defined as \(F = Z_1/d_1 / Z_2/d_2 = Z_1 / Z_2\) if \(d_1 = d_2 = 1\).
04

Compare \(T^2\) and \(F\) Definitions

Compare the \(T^2\) and \(F\) distribution definitions. \(T^2 = X^2 / Y\) matches the definition of a noncentral \(F\) distribution if both \(X^2\) and \(Y\) are chi-square with one degree of freedom. Hence, this shows that the square of the noncentral \(T\) random variable is a noncentral \(F\) random variable.

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Most popular questions from this chapter

Let \(\mathbf{A}=\left[a_{i j}\right]\) be a real symmetric matrix. Prove that \(\sum_{i} \sum_{j} a_{i j}^{2}\) is equal to the sum of the squares of the eigenvalues of \(\mathbf{A}\). Hint: If \(\boldsymbol{\Gamma}\) is an orthogonal matrix, show that \(\sum_{j} \sum_{i} a_{i j}^{2}=\operatorname{tr}\left(\mathbf{A}^{2}\right)=\operatorname{tr}\left(\mathbf{\Gamma}^{\prime} \mathbf{A}^{2} \mathbf{\Gamma}\right)=\) \(\operatorname{tr}\left[\left(\mathbf{\Gamma}^{\prime} \mathbf{A} \mathbf{\Gamma}\right)\left(\mathbf{\Gamma}^{\prime} \mathbf{A} \boldsymbol{\Gamma}\right)\right]\)

Let the independent normal random variables \(Y_{1}, Y_{2}, \ldots, Y_{n}\) have, respectively, the probability density functions \(N\left(\mu, \gamma^{2} x_{i}^{2}\right), i=1,2, \ldots, n\), where the given \(x_{1}, x_{2}, \ldots, x_{n}\) are not all equal and no one of which is zero. Discuss the test of the hypothesis \(H_{0}: \gamma=1, \mu\) unspecified, against all alternatives \(H_{1}: \gamma \neq 1, \mu\) unspecified.

Assume that the sample \(\left(x_{1}, Y_{1}\right), \ldots,\left(x_{n}, Y_{n}\right)\) follows the linear model \((9.6 .1)\). Suppose \(Y_{0}\) is a future observation at \(x=x_{0}-\bar{x}\) and we want to determine a predictive interval for it. Assume that the model \((9.6 .1)\) holds for \(Y_{0}\); i.e., \(Y_{0}\) has a \(N\left(\alpha+\beta\left(x_{0}-\bar{x}\right), \sigma^{2}\right)\) distribution. We will use \(\hat{\eta}_{0}\) of Exercise \(9.6 .4\) as our prediction of \(Y_{0}\) (a) Obtain the distribution of \(Y_{0}-\hat{\eta}_{0}\). Use the fact that the future observation \(Y_{0}\) is independent of the sample \(\left(x_{1}, Y_{1}\right), \ldots,\left(x_{n}, Y_{n}\right)\) (b) Determine a \(t\) -statistic with numerator \(Y_{0}-\hat{\eta}_{0}\). (c) Now beginning with \(1-\alpha=P\left[-t_{\alpha / 2, n-2}

Let \(X_{1}, X_{2}, X_{3}, X_{4}\) be a random sample of size \(n=4\) from the normal distribution \(N(0,1) .\) Show that \(\sum_{i=1}^{4}\left(X_{i}-\bar{X}\right)^{2}\) equals $$ \frac{\left(X_{1}-X_{2}\right)^{2}}{2}+\frac{\left[X_{3}-\left(X_{1}+X_{2}\right) / 2\right]^{2}}{3 / 2}+\frac{\left[X_{4}-\left(X_{1}+X_{2}+X_{3}\right) / 3\right]^{2}}{4 / 3} $$ and argue that these three terms are independent, each with a chi-square distribution with 1 degree of freedom.

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.
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