Question

Consider a random sample X1, ..., Xn from a normal distribution with known mean 0 and unknown variance 0 = 02 (a) Write the likelihood and log-likelihood function (b) Derive the maximum likelihood estimator for 6 (c) Show that the Fisher information matrix is I(O) = 2014 (d) What is the variance of the maximum likelihood estimator for @? Does it attain the Cramer-Rao lower bound? (e) Suppose that you are testing 0 = 1 versus the alternative 0 # 1. Determine the likeli- hood ratio test and the associated decision rule (f) Suppose that you are testing 0 = 1 versus the alternative 0 # 1. Determine the Wald test statistic and the associated decision rule (g) Suppose that you are testing 0 = 1 versus the alternative 0 # 1. Determine Rao's score statistic and the associated decision rule

Answer 1

Question @ - , Xn ~ N(0,02) ež(29)? -COL acco XI, X., bidofi, f(x) = OFK Likelihood function, L= 1 frui) Auswer log- likelihood function loge 2 = = - ndogo-n log 12a - zát Š xi? 2 ie loge? - A logo? - h log(27) - Xi 202 Fanswer Surice 0 =0?; logel can be written as, loge L = h logo log(24) 2 0 2 x ² 2 2 - n a loge do () х" -0 + 20 SWIT ㅇ 20 20 2

For M. L.E. of o, we must have a logel and & log! a logel 2 so gives - +20 Xi ao 2 2 le Xi ile = 20 202 20 M. L.E. of o 6 = 3x 2 Hence, = Xi Auswer

o from 0, a logel n 2 = 20 202 n 2 ² losh 2o + al- X 20 n E (delojes -ECxi ²) (0) 202 202 202 E(X)= 0 20 n (no) 202 n 202 02 202 2 20 E'loge ) Fisher Information = I fø) = ECOLD_) - E 121832) - ECOlond) - (0) Thus, Answer

@ s Wa n n M.LE. of 0 = ô = 5 x? Var (Ô) = Vas (xi?) Var (xi) E(Xi4) – {E(82);?] (364-662)2] EX~N10,62) var lo) - 204) = 204 Elxi +) = 364 1 = Elxi?) - 52 L = 204 Cramer - Rao lower bound = Ilo) n 204 204 Cramer - Pao lower bound. Thus MILE of D = Auswer