Question

Let X1, X2,..., Xn be a r.s. from f(x) = 0x0-1, for 0 < x <1,0 < a < 0o. (a) Find the MLE of 0. (b) Let T = -log X. Find the pdf of T. (c) Find the pdf of Y = DIT: (i.e., distribution of Y = - , log Xi). (d) Find E(). (e) Find E( ). (f) Show that the variance of 0 MLE → as n → 00. (g) Find the MME of 0.

Answer 1

sol: let X1, X3, . --Xn be a r.s from fra) = 0x01, < @ the likelihood fenelion of o based on random samffe 21, xa,...gdn is 7 L1012..","A) = n fixi) = 0"(axi)0"'. Ochill → Sog L1013*, --X) = niego +(0-1 & logize o google (01...m) = 2 + & logxi o L oe oog - (0124, - An) = -2 có Hoso - .: 9n maximum likelihood estimation method we morumbe Likelihood function wirit. barometer o. . to get the MLE af o we equate za Sogl (01X.-->$) = 0 and he log 2012,-X)<0 . foron 0 and 2 → z gogl 1018..) = 0 cod com Soy LC014,-)) <0. 3 + & populi 50 → ose - {MLE af e is Omie = © let TP-logy I calf of t is Ę14) = PITCH] = P[-logx <u) = P[dogx2-4) > F 4) = P(x), e-9) - 13 Elu) = i-e-ou p.def. cef T = -logy is 50cx< Fylus = d Flu) = ocou ; oculo

.: p.df. of To-Dogx is fylu) = jo e ou ; ocucco 1 0 ; ew © Ti, Tz,..., to are vid random vorable frin Flu) = o e-ou ; oculo en By Uniqueness theorem af migf (additive property of cramma distribution) y = 5 x nana (n, 9) 1 . p.dif. of Y = &Ti = - & log xi is Fy() = {21". Ynt e-by, ocycam o 'ew @ Elt) = 13 419dy M you coy oly F09 j yaitle=@ydly S put oy=t ody = dt => EL 4) = on Ligh" et it je - nė mota 371–

on Elta- ja hy en dry the god qoy day Team 5° you. c-0% duy con mentioned - po co tntlet dt 02 m2 > (614) = newney .nt2 2 -ñ.02 nanti) 02 © Verlance ( @meme) - tros on o and it is equal ! where I (0) is information 22 to Ile) = - El Page 1812,- o)) = Se Segue coment 23 = - · I(8) = - Elan bogli 014. Xn]) = 22 variance ( @mce) = zie) = 02 70 as moon - Vorcerere ( OMUC) = 2 40 as n700

to get the method of moment estimator of we equate sample moments to population moment. mor .: Elx) = f'xifte) de = SZ. 0x0! die = 0 $'z® dx = ( for mme of o.. we equate mi = Elx) ;mat Exi 0+1 ↑ î mio um, - o 011-mi) = m, 0 = mi Al-my ↑ os MME of ois Comme = m ; n = 1 &