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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 2

Question

Math 6010 • Linear Models 2005 • Question 2

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

(a) Find a test using the likelihood ratio.

(b) Provide a large sample approximation for the likelihood ratio test.

Raw LaTeX

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

(a) Find a test using the likelihood ratio.

(b) Provide a large sample approximation for the likelihood ratio test.

Topic Tags

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