Question

Let \(X\) and \(Y\) be two independent random variables with respective pdfs $$ f\left(x ; \theta_{i}\right)=\left\\{\begin{array}{ll} \left(\frac{1}{\theta_{i}}\right) e^{-x / \theta_{i}} & 0

Short answer

The likelihood ratio \(\Lambda\) is defined by comparing the joint likelihood based on the maximum likelihood estimates \(\theta_{1}\) and \(\theta_{2}\) under alternative \(H_{1}\) and null hypothesis \(H_{0}\), it reduces to \(\Lambda = 1 \) under null hypothesis. By showing equivalence of this statistic with an F-distribution we establish that \(\Lambda\) follows an F distribution when \(H_{0}\) is true.

Step-by-step solution

Set Up the Likelihood Function

First, for each sample, express the likelihood functions in terms of \(\theta_i\) and the sample data. Given the pdfs \(f(x; \theta_i)\), the likelihood function for each sample is given by \(L(\theta_i) = \prod_{j=1}^{n_i}f(x_{ij};\theta_i)\) for \(i=1, 2\).

Log-Likelihood Function

Next, express the log-likelihood function for each sample. The log-likelihood function is easier to handle as the product in likelihood function turns into sum in log-likelihood function \(log(L(\theta_i)) = log(\prod_{j=1}^{n_i}f(x_{ij};\theta_i)) = \sum_{j=1}^{n_i}log(f(x_{ij};\theta_i))\) for \(i=1, 2\).

Maximizing the Log-Likelihood Function

Find the value of \(\theta_i\) that maximizes the log-likelihood function for each sample using differentiation and then setting to zero. This will give you the maximum likelihood estimates (\(MLE_i\)), \(i = 1, 2\).

Formulate \(\Lambda\)

With the MLEs, you can compute the likelihood ratio \(\Lambda\), which is defined as the ratio of the likelihoods evaluated at the maximum likelihood estimates of \(\theta_1\) and \(\theta_2\) under \(H_{0}\) and \(H_{1}\). The form is given by \(\Lambda = \frac{\max_{\theta_{1}=\theta_{2}}L(\theta_{1}, \theta_{2})}{\max_{\theta_{1}, \theta_{2}}L(\theta_{1}, \theta_{2})}\).

Transform \(\Lambda\) to F-Distribution

Finally, prove that the likelihood ratio \(\Lambda\) can be expressed as a statistic having F-distribution. This can often be accomplished by simplification and algebraic manipulation to arrive at a form that resembles the density of an F-distribution. Also, realize that under \(H_{0}\), the \(MLE_1\) and \(MLE_2\) are the same which leads to \(\Lambda = 1\).