Question 1 Let X1, X2, .., Xn be a random sample drawn from a distribution with probability density function f (x; 0) = 7 ( 1 + x ) 0+ 1, 0 0 is an unknown parameter. (a) Show that Y, = In(1 + X;) follows a Gamma (a, b) distribution with probability density function ba g(v; a, b) = T ( a ) ya - de - by , y , a , b > 0 , and determine a and b. [4] (b) Construct the maximum likelihood estimator (MLE) of 0, namely 0. [4] (c) Is the MLE 0 unbiased? If not, is it asymptotically unbiased? [4] (d) Show that n202 Var (0) = (n - 1)2 (n - 2) Hence, show that O is a consistent estimator. [4] (e) Determine the asymptotic distribution of 0. [4] (f) Find a uniformly most powerful test for testing Ho: 0 = 1 and H1: 0 = 2 and express your rejection region with significance level a in terms of a quantile from a Chi-square distribution. At the 0.01 level of significance, what is your conclusion if the sample drawn is (0.6,0.6,0.8,0.8,0.9)? [4] (g) Find the uniformly most powerful test with significance level a for testing Ho: 0 = 1 and H1: 0 > 1. [4]