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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 \(\gamma^{2}\).

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Most popular questions from this chapter

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 $$ \sum_{1}^{n}\left(Y_{i}-\hat{\alpha}\right)^{2}=Q_{3}+\widehat{\beta}^{2} \sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} $$

Let \(\mu_{1}, \mu_{2}, \mu_{3}\) be, respectively, the means of three normal distributions with a common but unknown variance \(\sigma^{2}\). In order to test, at the \(\alpha=5\) percent significance level, the hypothesis \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}\) against all possible alternative hypotheses, we take an independent random sample of size 4 from each of these distributions. Determine whether we accept or reject \(H_{0}\) if the observed values from these three distributions are, respectively, $$ \begin{array}{lrrrr} X_{1}: & 5 & 9 & 6 & 8 \\ X_{2}: & 11 & 13 & 10 & 12 \\ X_{3}: & 10 & 6 & 9 & 9 \end{array} $$

Often in regression the mean of the random variable \(Y\) is a linear function of \(p\) -values \(x_{1}, x_{2}, \ldots, x_{p}\), say \(\beta_{1} x_{1}+\beta_{2} x_{2}+\cdots+\beta_{p} x_{p}\), where \(\boldsymbol{\beta}^{\prime}=\left(\beta_{1}, \beta_{2}, \ldots, \beta_{p}\right)\) are the regression coefficients. Suppose that \(n\) values, \(\boldsymbol{Y}^{\prime}=\left(Y_{1}, Y_{2}, \ldots, Y_{n}\right)\) are observed for the \(x\) -values in \(\boldsymbol{X}=\left[x_{i j}\right]\), where \(\boldsymbol{X}\) is an \(n \times p\) design matrix and its ith row is associated with \(Y_{i}, i=1,2, \ldots, n .\) Assume that \(Y\) is multivariate normal with mean \(\boldsymbol{X} \boldsymbol{\beta}\) and variance-covariance matrix \(\sigma^{2} \boldsymbol{I}\), where \(\boldsymbol{I}\) is the \(n \times n\) identity matrix. (a) Note that \(Y_{1}, Y_{2}, \ldots, Y_{n}\) are independent. Why? (b) Since \(\boldsymbol{Y}\) should approximately equal its mean \(\boldsymbol{X} \boldsymbol{\beta}\), we estimate \(\boldsymbol{\beta}\) by solving the normal equations \(\boldsymbol{X}^{\prime} \boldsymbol{Y}=\boldsymbol{X}^{\prime} \boldsymbol{X} \boldsymbol{\beta}\) for \(\boldsymbol{\beta}\). Assuming that \(\boldsymbol{X}^{\prime} \boldsymbol{X}\) is non- singular, solve the equations to get \(\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y}\). Show that \(\hat{\boldsymbol{\beta}}\) has a multivariate normal distribution with mean \(\boldsymbol{\beta}\) and variance-covariance matrix $$ \sigma^{2}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} $$ (c) Show that $$ (\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})^{\prime}(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})=(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta})^{\prime}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta})+(\boldsymbol{Y}-\boldsymbol{X} \hat{\boldsymbol{\beta}})^{\prime}(\boldsymbol{Y}-\boldsymbol{X} \hat{\boldsymbol{\beta}}) $$ say \(Q=Q_{1}+Q_{2}\) for convenience. (d) Show that \(Q_{1} / \sigma^{2}\) is \(\chi^{2}(p)\). (e) Show that \(Q_{1}\) and \(Q_{2}\) are independent. (f) Argue that \(Q_{2} / \sigma^{2}\) is \(\chi^{2}(n-p)\). (g) Find \(c\) so that \(c Q_{1} / Q_{2}\) has an \(F\) -distribution. (h) The fact that a value \(d\) can be found so that \(P\left(c Q_{1} / Q_{2} \leq d\right)=1-\alpha\) could be used to find a \(100(1-\alpha)\) percent confidence ellipsoid for \(\beta\). Explain.

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}

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a normal distribution \(N\left(\mu, \sigma^{2}\right)\). Show that $$ \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}=\sum_{i=2}^{n}\left(X_{i}-\bar{X}^{\prime}\right)^{2}+\frac{n-1}{n}\left(X_{1}-\bar{X}^{\prime}\right)^{2}, $$ where \(\bar{X}=\sum_{i=1}^{n} X_{i} / n\) and \(\bar{X}^{\prime}=\sum_{i=2}^{n} X_{i} /(n-1)\). Hint: \(\quad\) Replace \(X_{i}-\bar{X}\) by \(\left(X_{i}-\bar{X}^{\prime}\right)-\left(X_{1}-\bar{X}^{\prime}\right) / n\). Show that \(\sum_{i=2}^{n}\left(X_{i}-\bar{X}^{\prime}\right)^{2} / \sigma^{2}\) has a chi-square distribution with \(n-2\) degrees of freedom. Prove that the two terms in the right-hand member are independent. What then is the distribution of $$ \frac{[(n-1) / n]\left(X_{1}-\bar{X}^{\prime}\right)^{2}}{\sigma^{2}} ? $$

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