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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2009 • Question 5

Question

Math 6010 • Linear Models Spring 2009 • Question 5

Let \(X_1,\ldots ,X_n\) be a random sample from \(\operatorname {Poisson}(\theta )\).

(a) Find a complete, sufficient statistic.

(b) Noting that \(\Prob _\theta (X_1=1)=\theta e^{-\theta }\), use the Rao–Blackwell and Lehmann–Scheffé theorems to find a UMVUE of \(\tau (\theta ):=\theta e^{-\theta }\).

Raw LaTeX

Let \(X_1,\ldots,X_n\) be a random sample from \(\Poisson(\theta)\).

\textbf{(a)} Find a complete, sufficient statistic.

\textbf{(b)} Noting that \(\Prob_\theta(X_1=1)=\theta e^{-\theta}\), use the
Rao--Blackwell and Lehmann--Scheff\'e theorems to find a UMVUE of
\(\tau(\theta):=\theta e^{-\theta}\).

Topic Tags

Estimator Distributions Measure-Theoretic Probability Sufficiency, Completeness, and UMVUE Uniform Integrability and Truncation Methods