/*! 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 3 Let \(Y_{1}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 3

Let \(Y_{1}

Short Answer

Expert verified
Yes, \(4Y_{1}\), \(2Y_{2}\), and \(\frac{4}{3}Y_{3}\) are all unbiased estimators of \(\theta\). Their respective variances are \(0.2\theta^{2}\), \(0.22\theta^{2}\), and \(0.2\theta^{2}\).

Step by step solution

01

Prove unbiasedness

To prove that \(4Y_{1}\), \(2Y_{2}\), and \(\frac{4}{3}Y_{3}\) are unbiased estimators of \(\theta\), we have to show that their expected value \(E(\cdot)\) equals \(\theta\). That is, \(E(4Y_{1}) = E(2Y_{2}) = E(\frac{4}{3}Y_{3}) = \theta\). Recall that \(Y_{i}\) is the \({i}^{th}\) order statistic of a sample of size \(n\) from a uniform distribution \(U(0, \theta)\), and its distribution function is given by \(F_{Y_i}(y) = \binom{n}{i}(y/\theta)^i(1 - y/\theta)^{n-i}\), \(0 \leq y \leq \theta\).
02

Compute expected values

Now, let's calculate the expected values: \(E(Y_{1}) = \theta/4, E(Y_{2}) = \theta/2, E(Y_{3}) = 3\theta/4\). These are done by integrating the product of the order statistic \(y\) and its probability density function \(f(y)\), from \(0\) to \(\theta\), with respect to \(y\).
03

Validate unbiasedness

Substitute these values into the respective unbiased estimators: \(E(4Y_{1}) = 4E(Y_{1}) = 4(\theta/4) = \theta\), \(E(2Y_{2}) = 2E(Y_{2}) = 2(\theta/2) = \theta\), \(E(\frac{4}{3}Y_{3}) = \frac{4}{3}E(Y_{3}) = \frac{4}{3}(3\theta/4) = \theta\).Thus, these are indeed unbiased estimators of \(\theta\).
04

Compute variances

Now let鈥檚 compute variances for these estimators. For \(4Y_{1}\), \(2Y_{2}\), and \(\frac{4}{3}Y_{3}\), the variance formulas could be computed by computing \(Var(aX) = a^2Var(X)\), where \(a\) is a constant and \(Var(X)\) these variances are the variances of \(Y_1, Y_2, Y_3\) respectively. The variances of these variables can be computed by integrating the square of the difference of the variable and its expected value, multiplied by its probability density function, from 0 to \(\theta\), with respect to the variable.
05

Final Variances

After calculations, following variances are obtained: \(Var(4Y_{1}) = 16\theta^{2}/80 = 0.2\theta^{2}\), \(Var(2Y_{2}) = 4\theta^{2}/18 = 0.22\theta^{2}\), \(Var(\frac{4}{3}Y_{3}) = 16\theta^{2}/80 = 0.2\theta^{2}\).

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 \(Y_{1}

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pdf \(f(x ; \theta)=\theta^{2} x e^{-\theta x}, 00\) (a) Argue that \(Y=\sum_{1}^{n} X_{i}\) is a complete sufficient statistic for \(\theta\). (b) Compute \(E(1 / Y)\) and find the function of \(Y\) which is the unique MVUE of \(\theta\).

Show that the \(n\) th order statistic of a random sample of size \(n\) from the uniform distribution having pdf \(f(x ; \theta)=1 / \theta, 0

Let the pdf \(f\left(x ; \theta_{1}, \theta_{2}\right)\) be of the form $$ \exp \left[p_{1}\left(\theta_{1}, \theta_{2}\right) K_{1}(x)+p_{2}\left(\theta_{1}, \theta_{2}\right) K_{2}(x)+H(x)+q_{1}\left(\theta_{1}, \theta_{2}\right)\right], \quad a

In the preceding exercise, given that \(E(Y)=E[K(X)]=\theta\), prove that \(Y\) is \(N(\theta, 1)\) Hint: Consider \(M^{\prime}(0)=\theta\) and solve the resulting differential equation.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

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.