Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2008 • Question 8

Question

Math 6010 • Linear Models Spring 2008 • Question 8

Let \(X_i\) be a random sample of size \(n\) with pdf \(f(x;\theta )=\frac {2x}{\theta ^2}\) (\(0<x<\theta \)).

8a) (5 points) Use the MME \(\tilde \theta \) to find an unbiased estimator of \(\theta \).

8b) (10 points) Use the MLE \(\hat \theta \) to find another unbiased estimator of \(\theta \).

8c) (5 points) Using the factorization criterion, find one sufficient statistic for \(\theta \). Which of the two unbiased estimators you have found has a lower variance?

8d) (5 points) Calculate \(\operatorname {Var}(\tilde \theta )\) and \(\operatorname {Var}(\hat \theta )\). Does this confirm your answer?

Let \(X_i\) be a random sample of size \(n\) with pdf
\(f(x;\theta)=\frac{2x}{\theta^2}\) (\(0<x<\theta\)).

\textbf{8a)} (5 points) Use the MME \(\tilde\theta\) to find an unbiased
estimator of \(\theta\).

\textbf{8b)} (10 points) Use the MLE \(\hat\theta\) to find another unbiased
estimator of \(\theta\).

\textbf{8c)} (5 points) Using the factorization criterion, find one sufficient
statistic for \(\theta\). Which of the two unbiased estimators you have found
has a lower variance?

\textbf{8d)} (5 points) Calculate \(\Var(\tilde\theta)\) and \(\Var(\hat\theta)\).
Does this confirm your answer?

Topic Tags

Estimator Distributions Maximum Likelihood Estimation Probability Inequalities and Tail Bounds Sufficiency, Completeness, and UMVUE