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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 4

Question

Math 6010 • Linear Models Fall 2017 • Question 4

Let \(X_1,X_2,\ldots ,X_n\) be iid with density \[ h(t;\theta )=\frac {1}{2}I\{\theta -1\leq t\leq \theta +1\}. \] Find all maximum likelihood estimators of \(\theta \). Show that if \(\hat \theta _n\) is a maximum likelihood estimator then \(\mathbb {E}[\hat \theta _n]\to \theta \) as \(n\to \infty \).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be iid with density
\[
  h(t;\theta)=\frac{1}{2}I\{\theta-1\leq t\leq \theta+1\}.
\]
Find all maximum likelihood estimators of \(\theta\). Show that if
\(\hat\theta_n\) is a maximum likelihood estimator then
\(\E[\hat\theta_n]\to\theta\) as \(n\to\infty\).

Topic Tags

Estimator Distributions Likelihood Ratio Tests Maximum Likelihood Estimation Uniform Integrability and Truncation Methods