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 the independent random variables \(Y_{1}, Y_{2}, \ldots, Y_{n}\) have, respectively, the probability density functions \(N\left(\beta x_{i}, \gamma^{2} x_{i}^{2}\right), i=1,2, \ldots, n\), where the given numbers \(x_{1}, x_{2}, \ldots, x_{n}\) are not all equal and no one is zero. Find the maximum likelihood estimators of \(\beta\) and \(\gamma^{2}\).

Short Answer

Expert verified
By following the steps above, you will find the maximum likelihood estimators \(\hat{\beta}\) and \(\hat{\gamma}^{2}\) by setting the respective derivatives of the log-likelihood function to zero and solving them.

Step by step solution

01

Understanding the given information and determining the PDFs

We are given independent random variables \(Y_{1}, Y_{2}, \ldots, Y_{n}\) with respective probability density functions being normal distributions: \(N\left(\beta x_{i},\gamma^{2} x_{i}^{2}\right)\), for \(i=1,2, \ldots, n\). In the case of a normal distribution, the parameters \(\beta x_{i}\) and \(\gamma^{2} x_{i}^{2}\) represent the mean and the variance respectively.
02

Constructing the likelihood function

The likelihood function is the joint probability function of all the observations: \(L(\beta, \gamma^2) = \prod_{i=1}^{n} f_Y (y_i; \beta, \gamma^2)\). Since these observations are independent, the overall likelihood is the product of the individual likelihoods: \(L(\beta,\gamma^{2})=\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi\gamma^{2}x_{i}^{2}}}e^{\frac{-(y_{i}-\beta x_{i})^{2}}{2\gamma^{2}x_{i}^{2}}}\).
03

Taking the logarithm of the likelihood function

Take the natural logarithm of the likelihood function to simplify it. This step give us the log-Likelihood function: \( l(\beta, \gamma^2) = \ln L(\beta, \gamma^2) \).
04

Taking the derivatives of the log-likelihood function

Taking the derivative of the log-likelihood function with respect to \(\beta\) and setting it equal to zero gives the maximum likelihood estimator of \(\beta\). Similarly, taking the derivative of the log-likelihood function with respect to \(\gamma^{2}\) and setting it equal to zero gives the maximum likelihood estimator of \(\gamma^{2}\).

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

Let \(X_{i j k}, i=1, \ldots, a ; j=1, \ldots, b, k=1, \ldots, c\), be a random sample of size \(n=a b c\) from a normal distribution \(N\left(\mu, \sigma^{2}\right) .\) Let \(\bar{X}_{\ldots}=\sum_{k=1}^{c} \sum_{j=1}^{b} \sum_{i=1}^{a} X_{i j k} / n\) and $$ \begin{aligned} \bar{X}_{i_{r}} &=\sum_{k=1}^{c} \sum_{j=1}^{b} X_{i j k} / b c . \text { Prove that } \\ & \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{\ldots}\right)^{2}=\sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{i . .}\right)^{2}+b c \sum_{i=1}^{a}\left(\bar{X}_{i .}-\bar{X}_{\cdots}\right)^{2} \end{aligned} $$ Show that \(\sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{i . .}\right)^{2} / \sigma^{2}\) has a chi-square distribution with \(a(b c-1)\) degrees of freedom. Prove that the two terms in the right-hand member are independent. What, then, is the distribution of \(b c \sum_{i=1}^{a}\left(\bar{X}_{i .}-\bar{X}_{\ldots}\right)^{2} / \sigma^{2} ?\) Furthermore, let \(X_{. j .}=\sum_{k=1}^{c} \sum_{i=1}^{a} X_{i j k} / a c\) and \(\bar{X}_{i j .}=\sum_{k=1}^{c} X_{i j k} / c .\) Show that $$ \begin{aligned} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{\cdots}\right)^{2}=& \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{i j .}\right)^{2} \\ &+b c \sum_{i=1}^{a}\left(\bar{X}_{i_{n}}-\bar{X}_{\ldots}\right)^{2}+a c \sum_{j=1}^{b}\left(\bar{X}_{. j}-\bar{X}_{\ldots}\right)^{2} \\ &+c \sum_{i=1}^{a} \sum_{j=1}^{b}\left(\bar{X}_{i j .}-\bar{X}_{i .}-X_{. j .}+X_{\ldots}\right) \end{aligned} $$ Prove that the four terms in the right-hand member, when divided by \(\sigma^{2}\), are independent chi-square variables with \(a b(c-1), a-1, b-1\), and \((a-1)(b-1)\) degrees of freedom, respectively.

Suppose \(\mathbf{A}\) is a real symmetric matrix. If the eigenvalues of \(\mathbf{A}\) are only 0 's and 1 's then prove that \(\mathbf{A}\) is idempotent.

Let the independent random variables \(Y_{1}, \ldots, Y_{n}\) have the joint pdf. $$ L\left(\alpha, \beta, \sigma^{2}\right)=\left(\frac{1}{2 \pi \sigma^{2}}\right)^{n / 2} \exp \left\\{-\frac{1}{2 \sigma^{2}} \sum_{1}^{n}\left[y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}\right\\} $$ where the given numbers \(x_{1}, x_{2}, \ldots, x_{n}\) are not all equal. Let \(H_{0}: \beta=0(\alpha\) and \(\sigma^{2}\) unspecified). It is desired to use a likelihood ratio test to test \(H_{0}\) against all possible alternatives. Find \(\Lambda\) and see whether the test can be based on a familiar statistic. Hint: In the notation of this section show that $$ \sum_{1}^{n}\left(Y_{i}-\hat{\alpha}\right)^{2}=Q_{3}+\widehat{\beta}^{2} \sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} $$

Assume that the sample \(\left(x_{1}, Y_{1}\right), \ldots,\left(x_{n}, Y_{n}\right)\) follows the linear model \((9.6 .1)\). Suppose \(Y_{0}\) is a future observation at \(x=x_{0}-\bar{x}\) and we want to determine a predictive interval for it. Assume that the model \((9.6 .1)\) holds for \(Y_{0}\); i.e., \(Y_{0}\) has a \(N\left(\alpha+\beta\left(x_{0}-\bar{x}\right), \sigma^{2}\right)\) distribution. We will use \(\hat{\eta}_{0}\) of Exercise \(9.6 .4\) as our prediction of \(Y_{0}\) (a) Obtain the distribution of \(Y_{0}-\hat{\eta}_{0}\). Use the fact that the future observation \(Y_{0}\) is independent of the sample \(\left(x_{1}, Y_{1}\right), \ldots,\left(x_{n}, Y_{n}\right)\) (b) Determine a \(t\) -statistic with numerator \(Y_{0}-\hat{\eta}_{0}\). (c) Now beginning with \(1-\alpha=P\left[-t_{\alpha / 2, n-2}

The following are observations associated with independent random samples from three normal distributions having equal variances and respective means \(\mu_{1}, \mu_{2}, \mu_{3}\) $$ \begin{array}{rrr} \hline \text { I } & \text { II } & \text { III } \\ \hline 0.5 & 2.1 & 3.0 \\ 1.3 & 3.3 & 5.1 \\ -1.0 & 0.0 & 1.9 \\ 1.8 & 2.3 & 2.4 \\ & 2.5 & 4.2 \\ & & 4.1 \\ \hline \end{array} $$ Compute the \(F\) -statistic that is used to test \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3} .\)

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