Question

Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b) What is the MLE of p when it is known that py px p? - (c) Derive the likelihood ratio statistic for testing Ho : px px py PY vS HA (d) What is the asymptotic distribution of the likelihood ratio statistic when Ho is true? IMIM

Answer 1

* 01 PC1-R9 yol2 ohtained by Ihe MLE of 6lockhoad of a) findiong Cif Cun ti ating XXa, &,1 ith fepseet parametas and cauab on εφοδ like lehood of X, 1-Mi i-P h-2xi Σχ (4-1) ta kin g logasihm of both Side xilog t(n-i) -p) Diffeuntialiog Exi -A)(1) PX Zxi 0IMXI PA 1-PA -) Zxi n nu

2440 -pM) fe/pdz T 2Cy Pr 20-MYi tasing log, Loy (Hlg))= 21 Dyfirintiating Mgi t n-2yi (- Py Eyaating 20-2yi 1-Py Ir- Py R-M4i Σy( 20 -) Myi Py= whan PPP, likelib ed is ο Σχί 2n-29i n Σχί T aCyt (P) (I-P) HOx,y) P1-P)

xitzyi Eizyt) 120yi P 121 1-P) ong t12,9)lag 1cy Zy) lagP +230-2ityi) log (1-P) with especet to Diltuntiating Ot Eni 1291) 3n-EritZyi)1 1-P P 1-P Exi+yi CExi t Zyi) - ) Zxit Sgi P tot for toting PeRy 9atio The likelinoed seqp L(0))

Sapo when PrPy ats 1a boad ertro functin x,2 tng P. tolch nis is Log sup Lco) 30 xitzyi lilee lhbo d Jup LefO) functionof Jhm akd sepGHely When IXE log(x -2i log (- (Zyi)l tn - Tyi) log (1-Iy +Iogy t arn Z4n Enttzyi)log Exet y log Jap Lie +/Bn-Exitgi 30 30 Zyi -G0-1y1) 19(1-3 it Txityi

log Seap (0) Seap Lo) 3n ay log at 3-3y 101-5 NO og snt-hT) tog I- -gloy n-ng)lngT1) 7re ympto tic ais toribuhon lielhoed ahta I degsces offoedo with tst 3 Irat i3 yp oLlo)