Question

Consider the two uniform distributions with respective pdfs $$ f\left(x ; \theta_{i}\right)=\left\\{\begin{array}{ll} \frac{1}{2 \theta_{i}} & -\theta_{i}

Short answer

In this problem, the likelihood ratio statistic \(\Lambda\) used to test \(H0\) against \(H1\) is found to be 1. The distribution of \(-2 \log \Lambda\) when \(H0\) is true is 0. The degrees of freedom is 2.

Step-by-step solution

Identifying the Likelihood Function

For a random sample, the joint pdf is the product of the individual pdfs. The likelihood function \(L(\theta_1, \theta_2)\) is the joint pdf of the two samples viewed as a function of \(\theta_1\) and \(\theta_2\). Since both samples have the same number \(n_1 = n_2 = n\) for ease of the mathematics, the likelihood for \(H0:\theta_1 = \theta_2 =\theta\) is \(L(\theta) = 1/(2\theta)^{2n}\) if \(-\theta < X_1, Y_1 < \theta \) and 0 otherwise. While under the \(H1:\theta_1\neq \theta_2\), \(L(\theta_1, \theta_2) = 1/(2\theta_1)^n (2\theta_2)^n, -\theta_1 < X_1 < \theta_1, -\theta_2 < Y_1 < \theta_2\) and 0 otherwise.

Constructing the Likelihood Ratio

The likelihood ratio \(\Lambda\) is the ratio of maximum of the likelihood under \(H0\) to the maximum of the likelihood under \(H1\). Here, \(\Lambda = L(\theta)/L(\theta_1, \theta_2)\) where \(\theta\) maximizes the likelihood under \(H0\) and \(\theta_1\) and \(\theta_2\) maximize the likelihood under \(H1\). Maximizing the likelihood functions gives the maximum likelihood estimates \(\hat{\theta} = max{X_n, Y_n}\), \(\hat{\theta_1} = X_n, \hat{\theta_2} = Y_n\), substituting these estimates back into \(\Lambda\) gives \(\Lambda = (max{X_n, Y_n}/max{X_n, Y_n})^{2n} = 1.\)

Finding the distribution of \(-2 \log \Lambda\)

The distribution of \(-2 \log \Lambda\) when \(H0\) is true, where \(\Lambda = 1\), using the fact that \(\log (1) = 0\) and thus \(-2 \log \Lambda = 0\). This is a well-established result in statistics known as Wilk's theorem.

Understanding degrees of freedom.

Having found the likelihood ratio statistics, we need to find the distribution under Null hypothesis. Here, the dimensions of \(\Omega\) and \(\omega\) refers to the number of parameters to be estimated under the alternate and null hypotheses respectively. For this case, under \(H0\) there is 1 parameter, \(\theta\), to be estimated. But under \(H1\), there are 2 independent parameters, \(\theta_1 and \theta_2\), to be estimated. Thus, the number of degrees of freedom is 2 times the difference, which equals \(2*(2 - 1) = 2\).