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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 8

Question

Math 6010 • Linear Models Fall 2016 • Question 8

Let \(Y_1,Y_2,\ldots ,Y_n\) be independent random variables, each with the \(N(\beta x_i,\gamma ^2x_i^2)\) distribution, where the numbers \(x_1,x_2,\ldots ,x_n\) are known, not all are equal, and none of them is zero. Suppose \(\beta \) and \(\gamma \) are unknown. Find the maximum likelihood estimators of \(\beta \) and \(\gamma ^2\).

Raw LaTeX

Let \(Y_1,Y_2,\ldots,Y_n\) be independent random variables, each with
the \(\N(\beta x_i,\gamma^2x_i^2)\) distribution, where the numbers
\(x_1,x_2,\ldots,x_n\) are known, not all are equal, and none of them is zero.
Suppose \(\beta\) and \(\gamma\) are unknown. Find the maximum likelihood estimators
of \(\beta\) and \(\gamma^2\).

Topic Tags

Covariance and Independence Estimator Distributions Maximum Likelihood Estimation Uniform Integrability and Truncation Methods