Given that the probability density function is f(x)= theta*x^(theta-1)for testing parameter theta: Ho: theta = 1 and H1: theta = 2, the likelihood fuction is given by L= theta^2 * (x_1)^(theta-1) * (x_2)^(theta-1), the likelihood function under H1: L = 4(x_1)*(x_2), and the power of the test is 1-beta= .114 :
Find the most powerful test with significance level alpha = 5% to test Ho: theta = 2 against Ha: theta = 1.
How does the nature of the test statistic and rejection region for the most powerful test change if the Ha is changed to Ha: theta = 4.