Question

Let X, X,, ..., X, denote a random sample of size n from a population with pdf (10) = b exp(@m()).0<x<1 where (<O<0. Derive that the likelihood ratio test of H.:0=1 versus H, :0 #1 in terms of T(x) = ŽI (3)

Answer 1

Solutions we have to Given the values. X1, X2 ,---- Xn (Given) het XI,X2 .... an denote a random sample of size n from a population with probability density function. Given f(710) - EAP (100 in (^»), 05421 Where :OLOLO The likehood function for o=2 - 1 3 eap (10 In (-ai)) Exp , 10 (21) eco) = log < (o)=n logo + 1-0 En Incai) 11 os, decoration Ancai] =0 - 6 % sº Ja (4) Tes)

Guon the like hood ratio = 1 = SUPO zleco supo eco) I tip Lintnt (x)) Crain) = (RCX))Exp CNTCX )) Exp (n )" we reject to at the level a if ncc or; TCX )>k since CF (x ))" EXP Ent(x))exp and is decreasing function in T(X) and k is such that p (T (x) >KI Holan here, under HO IXIN uniform (0,1) then - -2 log xi NX2 hence. xi2 are indenpendent then under Ho, -2 error log Xi = 207 CX ) nxen

Then, The value of ' n likehood ratio PCT (*)> <IHO) =) P (2 NT (x)>20 K (HO) =) a =) ank =x20, 20 * * * * *