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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2008 • Question 4

Question

Math 6010 • Linear Models Spring 2008 • Question 4

Let \(X_i\) be a random sample of size \(n\) with pdf \(f(x;\theta )=\theta x^{\theta -1}\) (\(0<x<1\), \(\theta >0\).)

4a) (5 points) Find a complete and sufficient statistic.

4b) (5 points) Find a uniformly minimum variance unbiased estimator (UMVUE) of \(1/\theta \).

Raw LaTeX

Let \(X_i\) be a random sample of size \(n\) with pdf
\(f(x;\theta)=\theta x^{\theta-1}\) (\(0<x<1\), \(\theta>0\).)

\textbf{4a)} (5 points) Find a complete and sufficient statistic.

\textbf{4b)} (5 points) Find a uniformly minimum variance unbiased estimator
(UMVUE) of \(1/\theta\).

Topic Tags

Analysis of Variance Estimator Distributions Sufficiency, Completeness, and UMVUE Uniform Integrability and Truncation Methods