Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 4

Question

Math 6010 • Linear Models 2000 • Question 4

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

(a) Find the likelihood ratio test.

(b) Show that the likelihood ratio test is distribution free under \(H_0\).

(d) Approximate the critical values of the likelihood ratio test under \(H_0\).

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

(a) Find the likelihood ratio test.

(b) Show that the likelihood ratio test is distribution free under \(H_0\).

(c) Show that the likelihood ratio test is equivalent with the \(\chi^2\) test under \(H_0\).

(d) Approximate the critical values of the likelihood ratio test under \(H_0\).

Topic Tags

F-Tests and t-Tests Likelihood Ratio Tests Maximum Likelihood Estimation Quadratic Forms and Chi-Squared Distributions