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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 10

Question

Math 6010 • Linear Models 2005 • Question 10

Let \(\epsilon _i\), \(1\leq i\leq n\), be independent identically distributed normal \(N(0,\sigma ^2)\) random variables and consider the model \[ y_i=i\alpha +\gamma i^2+\epsilon _i,\qquad 1\leq i\leq n. \] We observe \(y_i\), \(1\leq i\leq n\).

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

(b) Compute the joint distribution of the maximum likelihood estimators for \((\alpha ,\gamma )\).

Raw LaTeX

Let \(\epsilon_i\), \(1\leq i\leq n\), be independent identically distributed normal \(N(0,\sigma^2)\) random variables and consider the model
\[
y_i=i\alpha+\gamma i^2+\epsilon_i,\qquad 1\leq i\leq n.
\]
We observe \(y_i\), \(1\leq i\leq n\).

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

(b) Compute the joint distribution of the maximum likelihood estimators for \((\alpha,\gamma)\).

Topic Tags

Covariance and Independence Estimator Distributions Maximum Likelihood Estimation Multivariate Normal Distributions