/*! 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 1 Let \(Y_{1}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 1

Let \(Y_{1}

Short Answer

Expert verified
A change of variables from \(Y_{i}\) to \(V_{1} = Y_{1} / Y_{4}\), \(V_{2} = (Y_{1}+Y_{2})/(Y_{3}+Y_{4})\), and \(V_{3} = Y_{4}\) and applying the Jacobian method yields the joint pdf. If the joint pdf disintegrates into the product of each individual pdfs, it demonstrates that the variables \(V_{1}\), \(V_{2}\), \(V_{3}\) are independent. Thus, it can be argued that the complete sufficient statistic \(Y_{4}\) for \(\theta\) is independent of each of the statistics \(Y_{1} / Y_{4}\) and \((Y_{1}+Y_{2})/\(Y_{3}+Y_{4}\).

Step by step solution

01

- Understand order statistics

The order statistics \(Y_{i}\) are just the observed values in increasing order. In this case, \(Y_{1}\) is the smallest observed value and \(Y_{4}\) is the largest.
02

- Find the joint pdf of order statistics

The joint pdf \(f(Y_{1}, Y_{2}, Y_{3}, Y_{4}; \theta)\) of the order statistics from a sample of size n=4 with uniform distribution on \((0, \theta)\) can be derived as \(f(y_{1}, y_{2}, y_{3}, y_{4}; \theta) = (n!) / \theta^{n},\), where \(0 < y_{1} < y_{2} < y_{3} < y_{4} < \theta\). After simplification, \((n!) / \theta^{n}\) translates to \(4/\theta^{4}\). For other combinations, the pdf is zero.
03

- Apply formula change to joint pdf

Apply a transformation of variables \(V_{1} = Y_{1} / Y_{4}\), \(V_{2} = (Y_{1}+Y_{2})/(Y_{3}+Y_{4})\), and \(V_{3} = Y_{4}\). The joint pdf of \(V_{1}\), \(V_{2}\), \(V_{3}\) can be calculated using Jacobian method.
04

- Prove independence of the variables

After calculating the joint pdf of \(V_{1}\), \(V_{2}\), \(V_{3}\), if the joint pdf can be disintegrated into the product of each individual pdf, then it shows that the variables \(V_{1}\), \(V_{2}\), \(V_{3}\) are independent.

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 the family of probability density functions \(\\{h(z ; \theta): \theta \in \Omega\\}\), where \(h(z ; \theta)=1 / \theta, 01\).

Let \(X_{1}, X_{2}, \ldots, X_{5}\) be iid with pdf \(f(x)=e^{-x}, 0

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a Poisson distribution with parameter \(\theta>0\) (a) Find the MVUE of \(P(X \leq 1)=(1+\theta) e^{-\theta}\). Hint: \(\quad\) Let \(u\left(x_{1}\right)=1, x_{1} \leq 1\), zero elsewhere, and find \(E\left[u\left(X_{1}\right) \mid Y=y\right]\), where \(Y=\sum_{1}^{n} X_{i}\) (b) Express the MVUE as a function of the mle of \(\theta\). (c) Determine the asymptotic distribution of the mle of \(\theta\). (d) Obtain the mle of \(P(X \leq 1)\). Then use Theorem \(5.2 .9\) to determine its asymptotic distribution.

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pmf \(p(x ; \theta)=\theta^{x}(1-\theta), x=0,1,2, \ldots\), zero elsewhere, where \(0 \leq \theta \leq 1\) (a) Find the mle, \(\hat{\theta}\), of \(\theta\). (b) Show that \(\sum_{1}^{n} X_{i}\) is a complete sufficient statistic for \(\theta\). (c) Determine the MVUE of \(\theta\).

Let \(Y_{1}
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

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.