/*! 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 7 Let \(X_{1}, X_{2}, \ldots, X_{n... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 7

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be iid \(N\left(\theta_{1}, \theta_{2}\right) .\) Show that the likelihood ratio principle for testing \(H_{0}: \theta_{2}=\theta_{2}^{\prime}\) specified, and \(\theta_{1}\) unspecified, against \(H_{1}: \theta_{2} \neq \theta_{2}^{\prime}, \theta_{1}\) unspecified, leads to a test that rejects when \(\sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} \leq c_{1}\) or \(\sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} \geq c_{2}\) where \(c_{1}

Short Answer

Expert verified
The problem solves the hypothesis testing problem using the likelihood ratio principle. The likelihood ratio test principle rejects the null hypothesis \( H_{0} \) when the sum of squared deviations, which is the maximum likelihood estimator of the variance under \( H_{0} \), is either smaller or larger than two selected constants \( c_{1} \) and \( c_{2} \) respectively, where \( c_{1} < c_{2} \).

Step by step solution

01

Understanding the likelihood ratio test principle

The likelihood ratio for a test of hypothesis \( H_{0} \) versus \( H_{1} \) is defined as: \( \Lambda = \frac{L(\theta_{2}^{\prime})}{L(\theta_{2})} \), where \( L(\theta) \) is the likelihood function which measures how likely the observed data are for different values of \( \theta \), and \( \theta \) and \( \theta' \) represent the parameter under \( H_{0} \) and \( H_{1} \) respectively.
02

Define the test statistic

Under the null hypothesis \( H_{0} \), the maximum likelihood estimator of the variance \( \sigma^2 \) of a normal distribution is the sample variance \( S^2 \), defined as \( S^2 = \sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} \). This will be our test statistic.
03

Determine the rejection regions

The Likelihood Ratio Test (LRT) tends to be most powerful when the likelihood under \( H_{1} \) is furthest from \( H_{0} \). Because of this, the LRT will tend to reject \( H_{0} \) when our test statistic is significantly greater or significantly less than what we would expect if the null hypothesis were true. This means that our rejection region will be when \( S^2 \leq c_{1} \) or \( S^2 \geq c_{2} \) where \( c_{1} < c_{2} \) are selected appropriately depending on the significance level of the test.

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

Consider a random sample \(X_{1}, X_{2}, \ldots, X_{n}\) from a distribution with pdf \(f(x ; \theta)=\theta(1-x)^{\theta-1}, 00\) (a) Find the form of the uniformly most powerful test of \(H_{0}: \theta=1\) against \(H_{1}: \theta>1\) (b) What is the likelihood ratio \(\Lambda\) for testing \(H_{0}: \theta=1\) against \(H_{1}: \theta \neq 1 ?\)

Let \(X_{1}, \ldots, X_{n}\) and \(Y_{1}, \ldots, Y_{m}\) follow the location model $$\begin{aligned} X_{i} &=\theta_{1}+Z_{i}, \quad i=1, \ldots, n \\ Y_{i} &=\theta_{2}+Z_{n+i}, \quad i=1, \ldots, m\end{aligned}$$ where \(Z_{1}, \ldots, Z_{n+m}\) are iid random variables with common pdf \(f(z)\). Assume that \(E\left(Z_{i}\right)=0\) and \(\operatorname{Var}\left(Z_{i}\right)=\theta_{3}<\infty\) (a) Show that \(E\left(X_{i}\right)=\theta_{1}, E\left(Y_{i}\right)=\theta_{2}\), and \(\operatorname{Var}\left(X_{i}\right)=\operatorname{Var}\left(Y_{i}\right)=\theta_{3}\). (b) Consider the hypotheses of Example \(8.3 .1\); i.e, $$H_{0}: \theta_{1}=\theta_{2} \text { versus } H_{1}: \theta_{1} \neq \theta_{2}$$ Show that under \(H_{0}\), the test statistic \(T\) given in expression \((8.3 .5)\) has a limiting \(N(0,1)\) distribution. (c) Using Part (b), determine the corresponding large sample test (decision rule) of \(H_{0}\) versus \(H_{1}\). (This shows that the test in Example \(8.3 .1\) is asymptotically correct.)

Let \(X_{1}, X_{2}, \ldots, X_{20}\) be a random sample of size 20 from a distribution which is \(N(\theta, 5) .\) Let \(L(\theta)\) represent the joint pdf of \(X_{1}, X_{2}, \ldots, X_{20} .\) The problem is to test \(H_{0}: \theta=1\) against \(H_{1}: \theta=0 .\) Thus \(\Omega=\\{\theta: \theta=0,1\\}\). (a) Show that \(L(1) / L(0) \leq k\) is equivalent to \(\bar{x} \leq c\). (b) Find \(c\) so that the significance level is \(\alpha=0.05 .\) Compute the power of this test if \(H_{1}\) is true. (c) If the loss function is such that \(\mathcal{L}(1,1)=\mathcal{L}(0,0)=0\) and \(\mathcal{L}(1,0)=\mathcal{L}(0,1)>0\), find the minimax test. Evaluate the power function of this test at the points \(\theta=1\) and \(\theta=0\)

If in Example \(8.2 .2\) of this section \(H_{0}: \theta=\theta^{\prime}\), where \(\theta^{\prime}\) is a fixed positive number, and \(H_{1}: \theta<\theta^{\prime}\), show that the set \(\left\\{\left(x_{1}, x_{2}, \ldots, x_{n}\right): \sum_{1}^{n} x_{i}^{2} \leq c\right\\}\) is a uniformly most powerful critical region for testing \(H_{0}\) against \(H_{1}\).

Suppose that a manufacturing process makes about 3 percent defective items, which is considered satisfactory for this particular product. The managers would like to decrease this to about 1 percent and clearly want to guard against a substantial increase, say to 5 percent. To monitor the process, periodically \(n=100\) items are taken and the number \(X\) of defectives counted. Assume that \(X\) is \(b(n=100, p=\theta)\). Based on a sequence \(X_{1}, X_{2}, \ldots, X_{m}, \ldots\), determine a sequential probability ratio test that tests \(H_{0}: \theta=0.01\) against \(H_{1}: \theta=0.05 .\) (Note that \(\theta=0.03\), the present level, is in between these two values.) Write this test in the form $$h_{0}>\sum_{i=1}^{m}\left(x_{i}-n d\right)>h_{1}$$ and determine \(d, h_{0}\), and \(h_{1}\) if \(\alpha_{a}=\beta_{a}=0.02\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.