Question

Let Yı, Y2, ...,Yn be an iid sample from a population distribution described by the pdf fy(y|0) = (@+ 1) yº, o<y<1 for 0> - -1. (a) Find the MOM estimator of 0. (b) Find the maximum likelihood estimator (MLE) of 0. (c) Find the MLE of the population mean E(Y) = 0 +1 0 + 2 You do not need to prove that the above is true. Just find its MLE.

Answer 1

Questions pidif, = 10+1 0 +2 012 Of / 0+2 Bry) = (0+1)yº ; ocje, The first population moment about Origui, Mi-Ely) y fry) dy= (y. (0+) yo dy = (0+)/ yot y=0 y=0 je w'= E(4) = 20 The frist Sample moment about origi, M'= ý = 5 ayi By Method of Moments, l'= M, (4) = 8 ie (@+1) = g(0+2) re. (C+1) = 0 ✓ + 2 ý i e. o(1-y)=29-1 649) (291 MOM Estimator of a Auswer iie ie Q- le = I-

16 bodig, f(y) = (0+1) yo ocycl likelihood function, L = flyi) (0+1)* ţ Yi n log (0+1) to log Yi i=1 log L a logL + log Yi do 0+1 For M. L.E. of o, we must have a los =o and lose co 202 alogh - O gives t logyi 0+! n 1 2:1 12 h loggi 0+ n 0+ logyi c=1 n n re OH! re O= -1 = & loggi CA logyi - 1 n M. L.E. of o logyi // Auswer

meau By Imvasiance Property of M.C.E. is obtained as, +) Ely) the M. L.E. of the populatisia ( Elogy )+1 = 0+2 -1 Elogy +2 :) logy) n Elogy) Elogyi-n Elogyi Elogy n-Elogy ( 11 Elogyi -| =( logyi = Auswer