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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 8

Question

Math 6010 • Linear Models 2000 • Question 8

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

(a) Find the maximum likelihood estimators for \(\alpha \) and \(\sigma ^2\).

(b) Find the distributions of the maximum likelihood estimators.

(c) Compute the least-squares estimator for \(\alpha \).

Raw LaTeX

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

(a) Find the maximum likelihood estimators for \(\alpha\) and \(\sigma^2\).

(b) Find the distributions of the maximum likelihood estimators.

(c) Compute the least-squares estimator for \(\alpha\).

Topic Tags

Estimator Distributions Generalized Least Squares Maximum Likelihood Estimation Multivariate Normal Distributions