Let \(X_1,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ f(t,\theta _1)= \begin {cases} 0, & \text {if } t\notin [1,\infty ),\\ \theta _1 t^{-\theta _1-1}, & \text {if } 1\leq t<\infty . \end {cases} \] Let \(Y_1,\ldots ,Y_m\) be independent, identically distributed random variables with density function \[ f(t,\theta _2)= \begin {cases} 0, & \text {if } t\notin [1,\infty ),\\ \theta _2 t^{-\theta _2-1}, & \text {if } 1\leq t<\infty . \end {cases} \] We wish to test \(H_0:\theta _1=\theta _2\) against the alternative that \(\theta _1\neq \theta _2\). The two samples are independent.
(a) Find a test using the likelihood ratio.
(b) Provide a large sample approximation for the likelihood ratio test.