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

The maximum likelihood estimators for \(\beta\) and \(\gamma^{2}\) are given by the solutions of the likelihood equations, derived from the log-likelihood after substituting the given density functions. Since no specific density functions were given, direct expressions for \(\beta\) and \(\gamma^{2}\) cannot be provided.

Step-by-step solution

Formulate the likelihood function

Given the probability density functions, we can form a likelihood function \(L(\beta, \gamma^{2})\) as the product of these density functions. Our likelihood function would be in the form \(L(\beta, \gamma^{2}) = \prod_{i=1}^{n} N\left(\beta x_{i}, \gamma^{2} x_{i}^{2}\right)\).

Take the natural logarithm of the likelihood function

To simplify the function, apply the log function on both sides, it helps turn products into sum and makes computations easier. Thus forming the log-likelihood function: \( l(\beta, \gamma^{2}) = \log (L(\beta, \gamma^{2})) = \sum_{i=1}^{n} \log(N\left(\beta x_{i}, \gamma^{2} x_{i}^{2}\right))\).

Find the partial derivatives of the log-likelihood function with respect to \(\beta\) and \(\gamma^{2}\)

By taking the derivative with respect to \(\beta\) and \(\gamma^{2}\), and setting these derivatives equal to zero, we can find the maximum likelihood estimators. This gives us a system of two equations, known as the likelihood equations: \(\frac{\partial l}{\partial \beta} = 0\) and \(\frac{\partial l}{\partial \gamma^{2}} = 0\).

Solve the likelihood equations

Solving the likelihood equations gives the solutions for \(\beta\) and \(\gamma^{2}\), which are the maximum likelihood estimators. Depending on the given density functions, expressions for \(\beta\) and \(\gamma^{2}\) will be obtained.\n