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

Let \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) be independent random samples from the two normal distributions \(N\left(0, \theta_{1}\right)\) and \(N\left(0, \theta_{2}\right)\). (a) Find the likelihood ratio \(\Lambda\) for testing the composite hypothesis \(H_{0}: \theta_{1}=\theta_{2}\) against the composite alternative \(H_{1}: \theta_{1} \neq \theta_{2}\). (b) This \(\Lambda\) is a function of what \(F\) -statistic that would actually be used in this test?

Short Answer

Expert verified
The likelihood ratio \(\Lambda\) for this hypothesis test is a function of the F-statistic, where the F-statistic is derived from the likelihood ratio using the formula \[F = \frac{n*m/2}{n+m-2} \times (1-\Lambda^{2/n})\].

Step by step solution

01

Setting Up the Likelihood Ratio

The likelihood ratio for the hypothesis test is defined as the maximized value of the likelihood function under the null hypothesis \(H_{0}: \theta_{1}=\theta_{2}\) divided by the maximized value of the likelihood function under the alternative hypothesis \(H_{1}: \theta_{1} \neq \theta_{2}\). In this case, the likelihood function for independent variables under a normal distribution is defined as the product of the individual probability density functions.
02

Express the Likelihood Ratio in terms of Sum of Squares

Transform the likelihood ratio by expressing it in terms of the sum of squares of the individual variables. This is achieved by replacing \(\theta_{1}\) and \(\theta_{2}\) in the formula by their estimates, which are the sum of squared deviations from the mean.
03

Deriving the F-statistic from the Likelihood Ratio

The relationship between the likelihood ratio (denoted by \(\Lambda\)) and the F-statistic is given by the formula -\[F = \frac{n*m/2}{n+m-2} \times (1-\Lambda^{2/n})\] Here, n and m are the sizes of the respective samples, \(\Lambda\) is the likelihood ratio.

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.

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

See all solutions

Recommended explanations on Math Textbooks

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