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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 3

Question

Math 6010 • Linear Models Fall 2016 • Question 3

Let \(X_1,X_2,\ldots ,X_n\) be iid with common pdf \[ f(x;\theta )=\frac {1}{2\theta }e^{-|x|/\theta }, \qquad -\infty <x<\infty , \] with \(\theta \) unknown but positive. Compute the Cramer-Rao lower bound for the variance of an unbiased estimator of \(\theta \), and extend your calculation to finding the UMVUE by finding a linear combination of \[ \sum _{i=1}^n \frac {\partial }{\partial \theta }\log f(X_i;\theta ) \] that is unbiased.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be iid with common pdf
\[
  f(x;\theta)=\frac{1}{2\theta}e^{-|x|/\theta},
  \qquad -\infty<x<\infty,
\]
with \(\theta\) unknown but positive. Compute the Cramer-Rao lower bound for the
variance of an unbiased estimator of \(\theta\), and extend your calculation to
finding the UMVUE by finding a linear combination of
\[
  \sum_{i=1}^n \frac{\partial}{\partial \theta}\log f(X_i;\theta)
\]
that is unbiased.

Topic Tags

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