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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 2

Question

Math 6010 • Linear Models Spring 2020 • Question 2

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with density function \[ f(t,\theta )= \begin {cases} 0, & \text {if } t\notin [-\theta ,\theta ],\\ \dfrac {5t^4}{2\theta ^5}, & \text {if } -\theta \leq t\leq \theta . \end {cases} \] Find the maximum likelihood estimator for \(\theta \) and compute its bias.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed
random variables with density function
\[
  f(t,\theta)=
  \begin{cases}
    0, & \text{if } t\notin[-\theta,\theta],\\
    \dfrac{5t^4}{2\theta^5}, & \text{if } -\theta\leq t\leq\theta.
  \end{cases}
\]
Find the maximum likelihood estimator for \(\theta\) and compute its bias.

Topic Tags

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