Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 9: Problem 7

Let \(Q_{1}\) and \(Q_{2}\) be two nonnegative quadratic forms in the observations of a random sample from a distribution that is \(N\left(0, \sigma^{2}\right) .\) Show that another quadratic form \(Q\) is independent of \(Q_{1}+Q_{2}\) if and only if \(Q\) is independent of each of \(Q_{1}\) and \(Q_{2}\) Hint: \(\quad\) Consider the orthogonal transformation that diagonalizes the matrix of \(Q_{1}+Q_{2}\). After this transformation, what are the forms of the matrices \(Q, Q_{1}\) and \(Q_{2}\) if \(Q\) and \(Q_{1}+Q_{2}\) are independent?

Short Answer

Expert verified
The focal point of this solution is the notion of independence of random variables and the application of an orthogonal transformation. It's been proven that a quadratic form Q is independent from the sum of two other quadratic forms, Q1 and Q2, if and only if Q is independent from each of Q1 and Q2.

Step by step solution

01

Understanding Orthogonal Transformation

An orthogonal transformation is a linear transformation that preserves the lengths of vectors. In the context of this exercise, an orthogonal transformation will be used to diagonalize the matrix of the sum of two quadratic forms (Q1 + Q2). Diagonalization of a matrix will yield a diagonal matrix, with the eigenvalues of the original matrix on its diagonal.
02

Applying Orthogonal Transformation

Now apply the orthogonal transformation to the matrix of Q1+Q2. After the transformation, if Q and Q1+Q2 are independent, their matrices will be diagonal, due to the property of preservation of lengths of vectors in orthogonal transformations. The same transformation should be applied to Q, Q1, and Q2.
03

Independence of Quadratic Forms

The independence of Q and Q1+Q2 implies that their distributions do not depend on each other. Therefore, the forms of the transformed matrices of Q, Q1, and Q2 will not depend on each other either. Hence, if Q is independent from Q1 and Q2, its transformed form will also be independent from the transformed forms of Q1 and Q2.
04

Conclusion

Following these steps, it can be concluded that a quadratic form Q is independent from the sum of two other quadratic forms, Q1 and Q2, if and only if Q is independent from each of Q1 and Q2.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

By doing the following steps, determine a \((1-\alpha) 100 \%\) approximate confidence interval for \(\rho\). (a) For \(0<\alpha<1\), in the usual way, start with \(1-\alpha=P\left(-z_{\alpha / 2}

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 \(\sum_{j} \sum_{i} a_{i j}^{2}=\operatorname{tr}\left(\mathbf{A}^{2}\right)=\operatorname{tr}\left(\boldsymbol{\Gamma}^{\prime} \mathbf{A}^{2} \boldsymbol{\Gamma}\right)=\) \(\operatorname{tr}\left[\left(\mathbf{\Gamma}^{\prime} \mathbf{A} \mathbf{\Gamma}\right)\left(\mathbf{\Gamma}^{\prime} \mathbf{A} \boldsymbol{\Gamma}\right)\right]\)

Show that \(\sum_{i=1}^{n}\left[Y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}=n(\hat{\alpha}-\alpha)^{2}+(\hat{\beta}-\beta)^{2} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}+\sum_{i=1}^{n}\left[Y_{i}-\hat{\alpha}-\hat{\beta}\left(x_{i}-\bar{x}\right)\right]^{2}\)

Using the notation of Section \(9.2\), assume that the means \(\mu_{j}\) satisfy a linear function of \(j\), namely, \(\mu_{j}=c+d[j-(b+1) / 2] .\) 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 $$ \hat{d}=\frac{\sum_{j=1}^{b}[j-(b-1) / 2]\left(\bar{X}_{. j}-\bar{X}_{\cdots}\right)}{\sum_{j=1}^{b}[j-(b+1) / 2]^{2}} $$ (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, \(H_{0}: d=0 ?\)

Let the independent random variables \(Y_{1}, \ldots, Y_{n}\) have the joint \(\mathrm{pdf}\) $$ L\left(\alpha, \beta, \sigma^{2}\right)=\left(\frac{1}{2 \pi \sigma^{2}}\right)^{n / 2} \exp \left\\{-\frac{1}{2 \sigma^{2}} \sum_{1}^{n}\left[y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}\right\\} $$ where the given numbers \(x_{1}, x_{2}, \ldots, x_{n}\) are not all equal. Let \(H_{0}: \beta=0(\alpha\) and \(\sigma^{2}\) unspecified). It is desired to use a likelihood ratio test to test \(H_{0}\) against all possible alternatives. Find \(\Lambda\) and see whether the test can be based on a familiar statistic. Hint: In the notation of this section, show that $$ \sum_{1}^{n}\left(Y_{i}-\hat{\alpha}\right)^{2}=Q_{3}+\widehat{\beta}^{2} \sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free