Question

I need d) only for a 2 parameter exponential defined (1/Theta)e^(-(x-n)/theta)).

Consider a random sample of size n from a two-parameter exponential distribution, X, EXPO, n), and let ñ and be the MLES. (a) Show that û and are independent. Hint: Use the results of Exercise 30 of Chapter 10. (b) Let V= 2n(8 – n)/0, V2 = 2rl – n/, and V, = 2n8/0. Show that V1 ~x?(2n), V3 ~x?(2) and V3 X (2n - 2). Hint: V1 = V2 + V3 and that V2 and V3 are independent. Find the MGF of V3 by the approach used in Theorem 8.3.6. (c) Show that (n-1) (-/ 0 F(2, 2n - 2). (d) Derive the GLR for a test of Horn = no versus Han> no. (e) Show that the critical region for a size a GLR test is equivalent to (1 - 1)(0-1/A> -12,2n - 2).

Answer 1

Sel" We know that CLR test for Hinn vs Hain no is given by following statistic. while do- parameter max Llon) space for to 2(x)= (e NERO max parameter Llon) 2a - Space forn, (O, NERA Rones, de a {(A, 2):870, OER} and ha= {(on): oo, no and on ER}. and Lion) represents the likelihood function of (@n). Senice, x xn is a random sample of size a from two parameter exponential distribution with pdf fla) I e de x>n, ner - (2-2) So, for dos {(0, No: 070, QER} the likelihood (20, 2) can be obtacred as - (220) L(0, 2) = le o I(2), X>No es Eixed constant. shvanatarantannaximigerthas, harnestihood It log-likelihood can be obtarna as. for maxima, log (8,0) = _ nloge - dolog (0%) = 0 2) + 2 log I (2; 2) + 1 = 2) +0 = 0. ) There fore, L(0,2) will be maximum when, 8 = 1 n and nar max s etelah 1(and); 8 = $ex=no) 40,0) = ône S12-20) (on) fro

A n for dea {(0, 2) 2 No; & so, n0ER, the likelihood function can be obtained a rin. LLOR) - 1 t.e & I (x;>"); Here, mle of o can be obtacred as obtained for Ho ine. ô - I (Xi-n). Beheren es MLE of n. 0 ñ for maxima over eet as rewrite the likelihood function as follows (-2) Lon= telt T. I n ). & X 7 min(x,xn) I(x ) So, if minimemis So, if minis greater than n Then all xi will be greater tha Heree, Lon) will be maximum for when ne is maximam. Herre, n < Ru, so, y es maximum when v=Rg. Therefore, the MLE fornis xp minst, arts). Also, if no un < R113 ) MLE for n = to. max Lcon- I po 2-2) where o IE (2 ) Contra An 11 -- zen reenee, 1 1 I Therefore, by Elleno) GLR test statistic is given by a - en I Lions en max 2 () @2) Esho = Regina max 400m sy X) TRT) contra awe ( (0-0) 1 an SWITWA