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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Summer 2020 • Question 5

Question

Math 6010 • Linear Models Summer 2020 • Question 5

Let \(X_1,X_2,\ldots ,X_n\) be independent and identically distributed random variables with density function \[ f(t)= \begin {cases} 0, & \text {if } -\infty <t<\theta ,\\ \dfrac {2\theta }{(\theta +t)^2}, & \text {if } \theta \leq t<\infty , \end {cases} \] for \(\theta >0\) a parameter. Find the maximum likelihood estimator for \(\theta \).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent and identically distributed
random variables with density function
\[
  f(t)=
  \begin{cases}
    0, & \text{if } -\infty<t<\theta,\\
    \dfrac{2\theta}{(\theta+t)^2}, & \text{if } \theta\leq t<\infty,
  \end{cases}
\]
for \(\theta>0\) a parameter. Find the maximum likelihood estimator for \(\theta\).

Topic Tags

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