Question

Problem 3. (30 pts) Let W, i = 1,...,n be iid Exp(6.), Vi, i = 1,...,m be iid Exp(02), and two samples are independent, fw(w) dhe , fv(u) = bene (c) Provide the MLE for (6,62) = (0 - 0). (d) If a UMVUE exists for (61 – 62), provide it; otherwise explain why it does not exist.

Answer 1

be ica Solution : 3 Let or (= lr dr. ...h he voi = 1,2..., m de Exp (Og) and two samples are independent, fw, (w) = t Fri) = the ulo, wyo va2 vyo The likeli beood flere lion for o, and (0,02% Til fagfors Bichos m Niller will I t e = T le [s их n wilco Vilar e m ah

og as 2. The log likelihood function for an aud loglos)= nlogo in logo Ilog L@,402) m vi ez wiwi nt t 5 261 6,2 ил Svi mt and Or we led 2 log109,163) den For MLE of a 2 204 > ท t al 6,2 o и И. 0 11 0,? Ewie w Ĝo, yo

For MLE of Oz we led 2 logb(0, 0) 209 мо mt a Evi S? =0 ņ ил Vi car М. 22 2 m Cantov Hereen the MLE of o andozas, @, = ū to and and = 0 t Swi ว่า เรา Evi mis m and T =

ovariance Let 7 (0,0z) = (9-) ? property of MLE wre qed the The MLE of 766.) as 2 TCG, 10) = (-5)? d We know theid) W; v Explo,) [=1,2,... The {(W;) = Cay Th E(162) = 802 det, T = ? Иu E(10,2 Thing E(Ti) = 34 240²-02 LE an

ni=1,2,.., u If Vin Exp (02) Then E(V.) = 9 E (1,2)= 26² Имо t Lot Tai Then E(T) = 2 ZmF M t Elv; 2) an amaz an = ² = Let 7(602) = (o; - Oz)? = (0,2+ Q ² - 20,0 Thens Ledo Torch) = 7+ Tg-awn

Hence the comvue of la paraula 7 (96) = ((-2)} & 2 7Q, CG1) where (Ti + 3-211) Ti - Inco T m au ir I Sviz an in Swi to = te and to the