Question

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

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

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.

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: \(L(\beta,\gamma^{2})=\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi\gamma^{2}x_{i}^{2}}}e^{\frac{-(y_{i}-\beta x_{i})^{2}}{2\gamma^{2}x_{i}^{2}}}\).

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: \( l(\beta, \gamma^2) = \ln L(\beta, \gamma^2) \).

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 \(\gamma^{2}\).