Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2015 • Question 1

Question

Math 6010 • Linear Models Fall 2015 • Question 1

Let \(X_1,X_2,\ldots ,X_n\) be independent, random variables with density functions \[ h(t;\theta _i)= \begin {cases} \dfrac {1}{\theta _i}e^{-t/\theta _i}, & \text {if } 0\leq t<\infty ,\\ 0, & \text {if } -\infty <t<0. \end {cases} \] We wish to test \(H_0:\theta _1=\theta _2=\cdots =\theta _n\) against the alternative that \(H_0\) is not true.

(a) Compute \(\lambda \), the generalized likelihood ratio and explain the application of the generalized likelihood test.

(b) Provide a large sample approximation for \(-2\log \lambda \) under \(H_0\).

Let \(X_1,X_2,\ldots,X_n\) be independent, random variables with
density functions
\[
h(t;\theta_i)=
\begin{cases}
\dfrac{1}{\theta_i}e^{-t/\theta_i}, & \text{if } 0\leq t<\infty,\\
0, & \text{if } -\infty<t<0.
\end{cases}
\]
We wish to test
\(H_0:\theta_1=\theta_2=\cdots=\theta_n\) against the alternative that \(H_0\) is
not true.

\textbf{(a)} Compute \(\lambda\), the generalized likelihood ratio and explain the
application of the generalized likelihood test.

\textbf{(b)} Provide a large sample approximation for \(-2\log\lambda\) under
\(H_0\).

Topic Tags

Covariance and Independence F-Tests and t-Tests Likelihood Ratio Tests Maximum Likelihood Estimation