Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 5

Question

Math 6010 • Linear Models Spring 2016 • Question 5

Let \(X_1,\ldots ,X_n\) be iid with the Gamma\((\theta ,\kappa )\) distribution, meaning that their common pdf is \[ f(x;\theta ,\kappa )=\frac {1}{\theta ^\kappa \Gamma (\kappa )} x^{\kappa -1}e^{-x/\theta }, \qquad x>0, \] with \(\kappa \) known but \(\theta \) unknown. Derive a \(100(1-\alpha )\%\) equal-tailed confidence interval for \(\theta \) based on the sufficient statistic.

Raw LaTeX

Let \(X_1,\ldots,X_n\) be iid with the Gamma\((\theta,\kappa)\)
distribution, meaning that their common pdf is
\[
  f(x;\theta,\kappa)=\frac{1}{\theta^\kappa\Gamma(\kappa)}
    x^{\kappa-1}e^{-x/\theta}, \qquad x>0,
\]
with \(\kappa\) known but \(\theta\) unknown. Derive a \(100(1-\alpha)\%\)
equal-tailed confidence interval for \(\theta\) based on the sufficient
statistic.

Topic Tags

Estimator Distributions Simultaneous Confidence Intervals Sufficiency, Completeness, and UMVUE Uniform Integrability and Truncation Methods