Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 7

Question

Math 6010 • Linear Models Spring 2016 • Question 7

Suppose that \(X_1,\ldots ,X_n\) are iid from a \(\BETA (\theta _1,\theta _2)\) distribution, meaning that the pdf is \[ f(x;\theta _1,\theta _2)= \frac {\Gamma (\theta _1+\theta _2)}{\Gamma (\theta _1)\Gamma (\theta _2)} x^{\theta _1-1}(1-x)^{\theta _2-1}, \qquad 0<x<1. \] Find jointly sufficient statistics for \(\theta _1\) and \(\theta _2\), and use them to find a UMVUEs for \(\theta _1\) and \(\theta _2\).

Raw LaTeX

Suppose that \(X_1,\ldots,X_n\) are iid from a
\(\BETA(\theta_1,\theta_2)\) distribution, meaning that the pdf is
\[
  f(x;\theta_1,\theta_2)=
  \frac{\Gamma(\theta_1+\theta_2)}{\Gamma(\theta_1)\Gamma(\theta_2)}
  x^{\theta_1-1}(1-x)^{\theta_2-1}, \qquad 0<x<1.
\]
Find jointly sufficient statistics for \(\theta_1\) and \(\theta_2\), and use them
to find a UMVUEs for \(\theta_1\) and \(\theta_2\).

Topic Tags

Estimator Distributions Multivariate Normal Distributions Sufficiency, Completeness, and UMVUE Uniform Integrability and Truncation Methods