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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 4

Question

Math 6010 • Linear Models Fall 2016 • Question 4

If a random sample of size \(n\) is taken from a distribution having pdf \(f(x;\theta )=2x/\theta ^2\), \(0<x\leq \theta \) and zero elsewhere, find:

(a) the MLE \(\hat \theta \) for \(\theta \),

(b) the constant \(c\) so that \(\mathbb {E}[c\hat \theta ]=\theta \),

(c) the MLE for the median of the distribution.

Raw LaTeX

If a random sample of size \(n\) is taken from a distribution having
pdf \(f(x;\theta)=2x/\theta^2\), \(0<x\leq\theta\) and zero elsewhere, find:

\textbf{(a)} the MLE \(\hat\theta\) for \(\theta\),

\textbf{(b)} the constant \(c\) so that \(\E[c\hat\theta]=\theta\),

\textbf{(c)} the MLE for the median of the distribution.

Topic Tags

Estimator Distributions Maximum Likelihood Estimation Multivariate Normal Distributions Uniform Integrability and Truncation Methods