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Chapter 9: Problem 7

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
The maximum likelihood estimators of \( \beta \) and \( \gamma^{2} \) are the solutions to the first order conditions obtained by differentiating the log-likelihood function.

Step by step solution

01

Form the log-likelihood function

Since we know that given \(Y_{i}\) follow a Normal distribution with mean \( \beta x_{i} \) and variance \( \gamma^{2}x_{i}^2 \), the probability density function can be written as: \(f_{Yi}(y) = \frac{1}{\sqrt{2\pi\gamma^2x_i^2}}e^{\frac{-(y-\beta x_i)^2}{2\gamma^2x_i^2}}\). The likelihood function is the product of the individual densities: \(L(\beta,\gamma^2 | y) = \prod_{i=1}^{n}f_{Yi}(y_i)\). Taking the logarithm, we arrive at the log-likelihood function.
02

Compute the partial derivatives

To find the MLE of \( \beta \) and \( \gamma^{2} \), we need to compute the derivative of the log-likelihood with respect to \( \beta \) and \( \gamma^{2} \), which gives the first order condition.
03

Solve the partial derivative equations

To find the maximum likelihood estimators, equate these partial derivatives to zero. The solutions for \( \beta \) and \( \gamma^{2} \) are the maximum likelihood estimators.

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Most popular questions from this chapter

The driver of a diesel-powered automobile decided to test the quality of three types of diesel fuel sold in the area based on mpg. Test the null hypothesis that the three means are equal using the following data. Make the usual assumptions and take \(\alpha=0.05\). \(\begin{array}{llllll}\text { Brand A: } & 38.7 & 39.2 & 40.1 & 38.9 & \\ \text { Brand B: } & 41.9 & 42.3 & 41.3 & & \\\ \text { Brand C: } & 40.8 & 41.2 & 39.5 & 38.9 & 40.3\end{array}\)

Show that \(\sum_{i=1}^{n}\left[Y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}=n(\hat{\alpha}-\alpha)^{2}+(\hat{\beta}-\beta)^{2} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}+\sum_{i=1}^{n}\left[Y_{i}-\hat{\alpha}-\hat{\beta}\left(x_{i}-\bar{x}\right)\right]^{2}\)

Suppose \(\boldsymbol{Y}\) is an \(n \times 1\) random vector, \(\boldsymbol{X}\) is an \(n \times p\) matrix of known constants of rank \(p\), and \(\boldsymbol{\beta}\) is a \(p \times 1\) vector of regression coefficients. Let \(\boldsymbol{Y}\) have a \(N\left(\boldsymbol{X} \boldsymbol{\beta}, \sigma^{2} \boldsymbol{I}\right)\) distribution. Discuss the joint pdf of \(\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y}\) and \(\boldsymbol{Y}^{\prime}\left[\boldsymbol{I}-\boldsymbol{X}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime}\right] \boldsymbol{Y} / \sigma^{2}\)

Let the \(4 \times 1\) matrix \(\boldsymbol{Y}\) be multivariate normal \(N\left(\boldsymbol{X} \boldsymbol{\beta}, \sigma^{2} \boldsymbol{I}\right)\), where the \(4 \times 3\) matrix \(\boldsymbol{X}\) equals $$ \boldsymbol{X}=\left[\begin{array}{rrr} 1 & 1 & 2 \\ 1 & -1 & 2 \\ 1 & 0 & -3 \\ 1 & 0 & -1 \end{array}\right] $$ and \(\boldsymbol{\beta}\) is the \(3 \times 1\) regression coefficient matrix. (a) Find the mean matrix and the covariance matrix of \(\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y}\). (b) If we observe \(\boldsymbol{Y}^{\prime}\) to be equal to \((6,1,11,3)\), compute \(\hat{\boldsymbol{\beta}}\).

Fit \(y=a+x\) to the data $$ \begin{array}{l|lll} \mathrm{x} & 0 & 1 & 2 \\ \hline \mathrm{y} & 1 & 3 & 4 \end{array} $$ by the method of least squares.
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