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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2015 • Question 8

Question

Math 6010 • Linear Models Spring 2015 • Question 8

(a) Let \(X_1,X_2,\ldots ,X_n\) be a random sample from \(N(\theta ,\theta ^2)\), where \(\theta >0\). Find a pair of statistics that are jointly sufficient.

(b) Using some linear combination of \(\sum _{i=1}^n X_i^2\) and \(\left (\sum _{i=1}^n X_i\right )^2\), show that the statistics of part (a) are not jointly complete.

Raw LaTeX

\textbf{(a)} Let \(X_1,X_2,\ldots,X_n\) be a random sample from
\(\N(\theta,\theta^2)\), where \(\theta>0\). Find a pair of statistics that are
jointly sufficient.

\textbf{(b)} Using some linear combination of \(\sum_{i=1}^n X_i^2\) and
\(\left(\sum_{i=1}^n X_i\right)^2\), show that the statistics of part (a) are not
jointly complete.

Topic Tags

Sufficiency, Completeness, and UMVUE Uniform Integrability and Truncation Methods