1. Let Y1, ..., Yn i.i.d following the exponential distribution, with pdf fly) = de-dy for y > 0. (a) [10 pts] Write down the likelihood function L(A) and the log likelihood function e(1) (b) [6 pts] Find the score function S(A) and the MLE for 1. (c) [6 pts] Find the Fisher information I(1). (d) [6 pts] The likelihood function satisfies the regularity conditions for the MLE to have asymptotic normality. Find the asymptotic distribution of the MLE. (e) [6 pts] Find the MLE for the log odds ratio 7 = log A and its asymptotic distribution, using your answer in (d). (f) [8 pts] Find the bias, variance and risk under squared error loss of the MLE of the mean n = 1/1. (g) [6 pts] Construct an a level Wald test for testing Ho : ) = do vs H1 : > * do (h) [6 pts] For testing Ho : ) = do vs H1 : A # do, construct an a level likelihood ratio test based on the x2 approximation. (i) [6 pts] Use the test you obtained in (h) to construct a 1 - a confidence interval of 1