Math 6010 • Linear Models

Consider the pdf \(f(x;\theta)=\theta x^{-(\theta+1)}\) (\(x\geq1\)).
Let \(X_i\) be a random sample of size \(n\) with pdf \(f(x;\theta_1)\). Let \(Y_i\)
be a random sample of size \(m\) with pdf \(f(x;\theta_2)\). The two sets of random
variables are independent. We wish to test
\[
  H_0:\theta_1=\theta_2
  \quad\text{against}\quad
  H_a:\theta_1\ne\theta_2.
\]

\textbf{6a)} (5 points) If one considers just the \(X\)-data, find the MLE
\(\hat\theta_1\).

\textbf{6b)} (5 points) Derive a formula for the critical region obtained
using the generalized likelihood ratio method.