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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2008 • Question 2

Question

Math 6010 • Linear Models Fall 2008 • Question 2

Let \(y_i=\alpha x_i+\epsilon _i\), \(1\leq i\leq n\), where \(\epsilon _1,\epsilon _2,\ldots ,\epsilon _n\) are independent identically distributed normal \(N(0,\sigma ^2)\) random variables.

a. Determine the maximum likelihood estimators for \(\alpha \) and \(\sigma ^2\). (5 points)

b. Compute the distribution of the maximum likelihood estimator for \(\alpha \). (5 points)

c. Compute the distribution of the maximum likelihood estimator for \(\sigma ^2\). (5 points)

Raw LaTeX

Let \(y_i=\alpha x_i+\epsilon_i\), \(1\leq i\leq n\), where
\(\epsilon_1,\epsilon_2,\ldots,\epsilon_n\) are independent identically
distributed normal \(N(0,\sigma^2)\) random variables.

a. Determine the maximum likelihood estimators for \(\alpha\) and \(\sigma^2\).
(5 points)

b. Compute the distribution of the maximum likelihood estimator for \(\alpha\).
(5 points)

c. Compute the distribution of the maximum likelihood estimator for
\(\sigma^2\). (5 points)

Topic Tags

Covariance and Independence Estimator Distributions Maximum Likelihood Estimation Multivariate Normal Distributions