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 \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a \(N\left(\theta_{1}, \theta_{2}\right)\) distribution. (a) Show that \(E\left[\left(X_{1}-\theta_{1}\right)^{4}\right]=3 \theta_{2}^{2}\). (b) Find the MVUE of \(3 \theta_{2}^{2}\).

Short Answer

Expert verified
For a normal distribution, the fourth central moment is calculated as \( E\left[\left(X_{1}-\theta_{1}\right)^{4}\right] = 3 \theta_{2}^{4} \). The MVUE (minimum variance unbiased estimator) of \( 3 \theta_{2}^{2} \) can be evaluated as \( 3s^2 \), where \( s^2 \) is sample variance.

Step by step solution

01

Identify the Moments of Normal Distribution

First, recognize that for a normally distributed random variable \(X \sim \mathcal{N}(\mu, \sigma^2)\), the fourth central moment, i.e., \(E\left[(X-\mu)^{4}\right] = 3 \sigma^{4}\).
02

Prove Given Equation

Substitute \( \mu = \theta_1 \) and \( \sigma = \theta_{2} \) into the equation we have, given that X is distributed normally with parameters \(\theta_{1}\) and \(\theta_{2}\) : \( E\left[\left(X_{1}-\theta_{1}\right)^{4}\right] = 3 \theta_{2}^{4} \).
03

Compute the MVUE of \(3 \theta_{2}^{2}\)

For point b), you should know that the minimum variance unbiased estimator (MVUE) of \( \sigma^2 = \theta_{2}^{2} \) in normal distribution is \( s^2 = \frac{1}{n-1} \sum_{i=1} ^{n} (X_i - \bar{X})^2 \), where \( \bar{X} = \frac{1}{n} \sum_{i=1} ^{n} X_i \). Following from the previous equation, for \( 3 \theta_{2}^{2} \), the MVUE would be \( 3s^2 \).

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 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

Prove that the sum of the observations of a random sample of size \(n\) from a Poisson distribution having parameter \(\theta, 0<\theta<\infty\), is a sufficient statistic for \(\theta\).

Let \(\left(X_{1}, Y_{1}\right),\left(X_{2}, Y_{2}\right), \ldots,\left(X_{n}, Y_{n}\right)\) denote a random sample of size \(n\) from a bivariate normal distribution with means \(\mu_{1}\) and \(\mu_{2}\), positive variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\), and correlation coefficient \(\rho .\) Show that \(\sum_{1}^{n} X_{i}, \sum_{1}^{n} Y_{i}, \sum_{1}^{n} X_{i}^{2}, \sum_{1}^{n} Y_{i}^{2}\), and \(\sum_{1}^{n} X_{i} Y_{i}\) are joint complete sufficient statistics for the five parameters. Are \(\bar{X}=\) \(\sum_{1}^{n} X_{i} / n, \bar{Y}=\sum_{1}^{n} Y_{i} / n, S_{1}^{2}=\sum_{1}^{n}\left(X_{i}-\bar{X}\right)^{2} /(n-1), S_{2}^{2}=\sum_{1}^{n}\left(Y_{i}-\bar{Y}\right)^{2} /(n-1)\), and \(\sum_{1}^{n}\left(X_{i}-\bar{X}\right)\left(Y_{i}-\bar{Y}\right) /(n-1) S_{1} S_{2}\) also joint complete sufficient statistics for these parameters?

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be iid with the distribution \(N\left(\theta, \sigma^{2}\right),-\infty<\theta<\infty\). Prove that a necessary and sufficient condition that the statistics \(Z=\sum_{1}^{n} a_{i} X_{i}\) and \(Y=\sum_{1}^{n} X_{i}\), a complete sufficient statistic for \(\theta\), are independent is that \(\sum_{1}^{n} a_{i}=0 .\)

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be iid \(N(0, \theta), 0<\theta<\infty\) Show that \(\sum_{1}^{n} X_{i}^{2}\) is a sufficient statistic for \(\theta\).

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from \(N\left(\theta_{1}, \theta_{2}\right)\). (a) If the constant \(b\) is defined by the equation \(P(X \leq b)=0.90\), find the mle and the MVUE of \(b\). (b) If \(c\) is a given constant, find the mle and the MVUE of \(P(X \leq c)\).

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