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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2018 • Question 7

Question

Math 6010 • Linear Models Fall 2018 • Question 7

Suppose that \(X_1,\ldots ,X_n\) form a random sample from a distribution for which the pdf \(f(x|\theta _1,\theta _2)\) is as follows: \[ f(x|\theta _1,\theta _2)= \begin {cases} \dfrac {1}{\theta _2-\theta _1}, & \theta _1<x<\theta _2,\\ 0, & \text {otherwise.} \end {cases} \] The values of \(\theta _1\) and \(\theta _2\) are unknown, other than \(-\infty <\theta _1<\theta _2<\infty \). Find their MLEs.

Raw LaTeX

Suppose that \(X_1,\ldots,X_n\) form a random sample from a
distribution for which the pdf \(f(x|\theta_1,\theta_2)\) is as follows:
\[
  f(x|\theta_1,\theta_2)=
  \begin{cases}
    \dfrac{1}{\theta_2-\theta_1}, & \theta_1<x<\theta_2,\\
    0, & \text{otherwise.}
  \end{cases}
\]
The values of \(\theta_1\) and \(\theta_2\) are unknown, other than
\(-\infty<\theta_1<\theta_2<\infty\). Find their MLEs.

Topic Tags

Estimator Distributions Multivariate Normal Distributions Quadratic Forms and Chi-Squared Distributions Uniform Integrability and Truncation Methods