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5. (10 marks) Let Y, . . ., Y,, denote a random sample from the probability density function f (3; 0) - - exp _(logy/)? ] for y > 0 and 0 > 0 20 and f(y; 0) = 0 otherwise. You may assume that for this distribution E(log Y,) = 0 and Var(log Y) = 0. (a) Find the maximum likelihood estimator (MLE) of 0. (b) Find the expected information and provide an approximate standard error for the MLE in (a). (c) Consider testing the hypothesis that the parameter Of is equal to a particular value. Ho : 0 = 1 versus H1 : 0 / 1. Write down the test statistics corresponding to each of the following tests and evaluate them under the assumption that n = 100 and the observed value of the MLE is 0 = 1.2: i. likelihood ratio test ii. score test iii. Wald test iv. approximate Wald test