Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2008 • Question 4

Question

Math 6010 • Linear Models Fall 2008 • Question 4

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ f(t;\theta )= \begin {cases} 0, & \text {if } t<\theta ,\\ e^{-(t-\theta )}, & \text {if } t>\theta . \end {cases} \]

a. Find a moment estimator for \(\theta \). (5 points)

b. Find the maximum likelihood estimator for \(\theta \). (5 points)

c. Determine the asymptotic efficiency between the two estimators. (5 points)

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent, identically distributed
random variables with density function
\[
  f(t;\theta)=
  \begin{cases}
    0, & \text{if } t<\theta,\\
    e^{-(t-\theta)}, & \text{if } t>\theta.
  \end{cases}
\]

a. Find a moment estimator for \(\theta\). (5 points)

b. Find the maximum likelihood estimator for \(\theta\). (5 points)

c. Determine the asymptotic efficiency between the two estimators. (5 points)

Topic Tags

Covariance and Independence Estimator Distributions Maximum Likelihood Estimation Uniform Integrability and Truncation Methods