/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 If \(X_{1}, X_{2}, \ldots, X_{n}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 8

If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from a beta distribution with parameters \(\alpha=\beta=\theta>0\), find a best critical region for testing \(H_{0}: \theta=1\) against \(H_{1}: \theta=2\)

Short Answer

Expert verified
The best critical region \(C = \{ x: \lambda \le k \}\) is determined by calculating the likelihood ratio and setting up criteria based on the level of significance \(\alpha\). Here, \(k\) is a constant determined to ensure that the size of the critical region is equal to the given level of significance \(\alpha\).

Step by step solution

01

Definition of Beta Distribution

The probability density function of the Beta distribution is given by: \[f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\] for \(0 \le x \le 1\) and \(\alpha > 0\), \(\beta > 0\). Where \(B(\alpha,\beta)\) stands for the beta function.
02

Calculate the Likelihood ratio

The likelihood ratio for two simple hypotheses \(H_{0}: \theta=\theta_{0}\) and \(H_{1}: \theta=\theta_{1}\) is computed as: \[ \lambda = \frac{L(\theta_{0})}{L(\theta_{1})}\] where \(L(\theta)\) is the likelihood function. In this case, \(\theta_{0} = 1\), \(\theta_{1} = 2\), hence \[ \lambda = \frac{L(1)}{L(2)} \] Calculate the likelihood ratios by replacing all \(\theta\) with 1 in \(H_{0}\) and 2 in \(H_{1}\) in the likelihood function derived from the Beta distribution.
03

Define the Best Critical Region

Define the best critical region (rejection region) as: \[C = \{ x: \lambda \le k \}\] for some constant \(k > 0\). In this step, find the value of \(k\) such that the size of the critical region is equal to the given level of significance \(\alpha\). Typically, \(\alpha\) is set to be 0.05.

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 91影视!

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

Let the random variable \(X\) have the pdf \(f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a normal distribution \(N(\theta, 16)\). Find the sample size \(n\) and a uniformly most powerful test of \(H_{0}: \theta=25\) against \(H_{1}: \theta<25\) with power function \(\gamma(\theta)\) so that approximately \(\gamma(25)=0.10\) and \(\gamma(23)=0.90\).

Let \(X_{1}, \ldots, X_{n}\) denote a random sample from a gamma-type distribution with \(\alpha=2\) and \(\beta=\theta .\) Let \(H_{0}: \theta=1\) and \(H_{1}: \theta>1\) (a) Show that there exists a uniformly most powerful test for \(H_{0}\) against \(H_{1}\), determine the statistic \(Y\) upon which the test may be based, and indicate the nature of the best critical region. (b) Find the pdf of the statistic \(Y\) in Part (a). If we want a significance level of \(0.05\), write an equation which can be used to determine the critical region. Let \(\gamma(\theta), \theta \geq 1\), be the power function of the test. Express the power function as an integral.

Let \(\left(X_{1}, Y_{1}\right),\left(X_{2}, Y_{2}\right), \ldots,\left(X_{n}, Y_{n}\right)\) be a random sample from a bivariate normal distribution with \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}, \rho=\frac{1}{2}\), where \(\mu_{1}, \mu_{2}\), and \(\sigma^{2}>0\) are unknown real numbers. Find the likelihood ratio \(\Lambda\) for testing \(H_{0}: \mu_{1}=\mu_{2}=0, \sigma^{2}\) unknown against all alternatives. The likelihood ratio \(\Lambda\) is a function of what statistic that has a well- known distribution?

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a normal distribution \(N(\theta, 100) .\) Show that \(C=\left\\{\left(x_{1}, x_{2}, \ldots, x_{n}\right): c \leq \bar{x}=\sum_{1}^{n} x_{i} / n\right\\}\) is a best critical region for testing \(H_{0}: \theta=75\) against \(H_{1}: \theta=78\). Find \(n\) and \(c\) so that $$P_{H_{0}}\left[\left(X_{1}, X_{2}, \ldots, X_{n}\right) \in C\right]=P_{H_{0}}(\bar{X} \geq c)=0.05$$ and $$P_{H_{1}}\left[\left(X_{1}, X_{2}, \ldots, X_{n}\right) \in C\right]=P_{H_{1}}(\bar{X} \geq c)=0.90$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.