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UNlVERSlTl SAINS MALAYSIA First Semester Examination Academic Session 2004/2005 October 2004 MST 561 - STATISTICAL INFERENCE [PENTAABIRAN STATISTIK] Duration : 3 hours [Masa : 3 jam] Please check that this examination paper consists of MINE [9] pages of printed material before you begin the examination. [Sila pastikan bahawa kertas peperiksaan ini mengandungi SEMBILAN [9] muka surat yang bercetak sebelum anda memulakan peperiksaan ini.] Answer all FIVE questions. Jawab semua LIMA soalan. * . .2/- 2 [MST 5611 1. (a) Let X,, X, ,..., X, be independent, identically distributed random variables with common probability density function. Let = XCl,,Y2 = X(,),...,q = X,,, be the order statistics for Let 2, = nT,Z,  = (n  -l)(Y2 - q )  ,..., 2, = (Y, -q-]). Show that 2, , 2, , ...) 2, are independent and identically distributed random variables. [50 marks] X2 ,**.> x n  * (b) Let X, ,X ,  ,..., X ,  be a random sample of size n fiom a Poisson distribution with parameter A = 1, i.e. , x=O,1,2 ,..., i= l , 2  ,..., n e-l P(Xi  = x )  = - X !  (i) Derive the m.g.f. for X, .X I  +X2 +.A x, (ii) Let F,, = . Consider n Y, = &(Z, -1). Show that the m.g.f for Yn is given by exp [- t& + n (er'& - l ) ]  Hence, obtain the limiting distribution of Y, when n + 00. (iii) [50 marks] 1. (a) Katakan XI, X, ,..., X ,  adalah pembolehubah-pembolehubah rawak takbersandar dan tertabur secaman dengan Ji-ngsi ketumpatan kebarangkalian sepunya Katakan q = X o ) ,  Y, = X(z ,  )..., Y, = X,,, adalah statistik tertib bagi XI, X, ,. .., X, . Katakan 2, = n q  ,Z, = (n  -l)(Y2 - q )  ,..., 2, = (Y, - Y,-,). Tunjukkan bahawa 2, ,Z, ,..., Z,  adalah pembolehubah-pembolehubah rawak takbersandar dan tertabur secaman. [SO markah] (3) Katakan X, ,X z  ,..., X, satu sampel rawak saiz n daripada suatu taburan Poisson dengan parameter R = 1, iaitu, P(Xi  = x )  = - e-' , x=O,1,2 ,..., i=1,2). . . ,n X !  . . .3/- 3 [MST 5611 (i) TerbitkanJp.m bagi X i .  (ii) Biarkan 2, = n x,+x, t...+xn Pertimbangkan Y, = &(z, - 1). Tunjukkan f .p.m b agi Y, d iberi oleh exp [- t& + n (et'& - 1)) (iii) Demikian, dapatkan taburan penghad bagi Yn apabila n + C O .  [50 markah] 2. (a) Let Y; < Yz < & < Y4 be the order statistics fkom a random sample of size 4 taken from a distribution with p.d.f. 2 x ,  O<x<l  0 , elsewhere' Obtain the joint p.d.f. of & and Y,. Find the conditional p.d.f. of 5 given Y4 = y4  Evaluate E(Y, I y,) . (i) (ii) (iii) [30 marks] (b) State and prove Chebychev's inequality. [ 3 0 marks] (c) Let X,, X, ,..., X, be a random sample from a distribution with p.d.f. (i) (ii) Find the MLE for 8. Find the MLE for P, {XI - 8 2 l} [40 marks] 2. (a) Katakan Y; < Yz < 5 < Y4 statistik tertib daripada suatu sampel rawak saiz n = 4yang diambil daripada suatu taburan denganjk.k. 2 x ,  O<x<l 0 , di tempat lain' Dapatkan$k.k. tercantum bagi Y, dan Y,. Cari$k.k. bersyarat bagi Y3 diberi Y4 = y,. (ij (ii) (iii) Nilaikan E(& I y,) .  [30 markah] . . AI- 4 [MST 5611 (E) Nyatakan dan buktikan kztaksamaan Chebychev. [30 markah] (c) Katakan X, ,X, ,..., X ,  adalah suatu sampel rawak darijbngsi kztumpatan kebarangkalian (i) Carikan MLE 8.  (ii) Carikan MLE bagi PO {X, -0 2 1). [40 markah] 3. (a) What are the roles of the following theorems in estimation theory: (i) (ii) Rao - Blackwell Theorem. (iii) Lehmann - Scheffe Theorem. (iv) Cramer - Rao Theorem? Give examples to illustrate your answer. Neyman - Fisher Factorization Theorem [40 marks] (b) Let XI , X, ,...,X, be a random sample from a distribution with p.d.f h - h ,  x>o , i l >o  0 , elsewhere n (i) Show that T = for A . X i  is a statistic which is suflicient and complete i=1 O i f X ,  < K ,  1 i f  X, 2 K ,  (ii) Let V ( X , ) =  where K is a known constant. Show thanU(X,)is unbiased for e-KA. Hence, show that the uniformly minimum variance unbiased estimator for e-KR is given by (iii) n-1 when T 2 K  when T < K  [60 marks] 3. (a) Apakah peranan teorem-teorem berikut di dalam teori penganggaran: (0 Teorem Pemfaktoran Neyman-Fisher, (ii) Teorem Rao-Blackwell, (iii) Teorem Lehmann-Schefe, (iv) Teorem Cramer-Rao? Beri contoh-contoh untuk mengilustrasi jawapan anda. [40 markah] . . .5l- [MST 5611 5 (b) Katakan X ,  , X, ,..., X ,  suatu sampel rawak daripada suatu taburan (0 Tunjukkan T =c X i  ialah suatu statistik yang cukup dan n i-1 lengkap bagi A. 0 jika X, < K ,  1 jikaX, > K ,  (ii) Katakan V(Xl)= . -  di mana K ialah suatu pemalar yang diketahui. Tunjukkan U(X,)sahama bagi e-'*. Demikian, tunjuhn bahawa penganggar sahama bewarians minimum secara seragam bagi e-KA diberi oleh (iii) apabila T 2 K apabila T < K \ [60 markah] 4. (a) Let X, ,X, ,..., X, be a random sample fiom a distribution with p.d.f. ( p - 1 ,  O<X<1,0>0 0 , elsewhere . Suppose than the prior distribution for 0 is given by the p.d.f. 0 r -1 -8% m=[ r ( r )  0 > 0  where r and il are known positive constarts. 0, elsewhere (i) (ii) Find the posterior p.d.f. for 0 Find the Bayes estimator for 0 with respect to the prior n(B) above using the squared error loss function [40 marks] (b) Let XI, X, ,..., Xn be a random sample of size from a distribution with p.d.f. e(i+x)-(l+8),o<x<03,e>o 0 , elsewhere 1 (i) (ii) Find the m.1.e. for (a) 0 and (b) -. eDoes there exist a complete and sufficient statistic for 8? . . .6l- 6 [MST 5611 1 Find a uniformly minimum variance unbiased estimator for (a) - (iii) e (iv) Find the Cramer-Rao lower bound for the variances of unbiased estimators of -. 1 e [60 marks] 4. (a) Katakan X, X, ,...) X ,  suatu sampel rawak daripada suatu taburan denganJk.k. Andaikan bahawa taburan priori bagi 0 diberi o1ehJk.k. x , g r - l e - B A  r ( r )  8 > 0  di rnana r and A adalah pemalar positifyang diketahui. 0, ditempatlain (i) CariJk.k. posterior bagi 0 (ii) Cari penganggar Bayes bagi 0 terhadap prior z(e) di atas dengan menggunakan fungsi kerugian ralat kuasa dua. [40 markuh] (b) Katakan X, X, ,..., X ,  suatu sampel rawak saiz n daripada suatu taburan dengan Jk.k. 1 Carip.k.m bagi (a) 0 dan 0) -. Adakah wujudnya suatu statistik yang mkup dan lengkap bagi 8 ? seragam bagi (a) - dan (a) 8. Cari batas bawah Cramer-Rao bagi varians penganggar sahama (i) (ii) (iii) Cari suatu penganggar saksama bervarians minimum secara e 1 e (iv) 1 untuk - 8 '  [60 markah] 5. (a) State the Neyman-Pearson fundamental lemma. [ 10 marks] . . .?I- [MST 5611 a (b) Describe the concepts which are relevant in the theory of hypothesis testing . [20 marks] (c) Let X, ,X, be a random sample of size 2 from a distribution with p.d.f. (l+e)Xs ,o I x I i,e > o 0 , elsewhere Show that the uniformly most powerful test for testing H ,  : B = 2 versus H, :9=8, € 2  isgivenby What is the size of the above test? [ 3 0 marks] (d) Let X,, X, ,..., X, be a random sample from a distribution with p.d.f. Let & ,Y, )...) Y, be a random sample from a distribution with p.d.f. Suppose that the two samples are independent. Show that the likelihood ratio test for testing H,  :U’ = r 2 , p  andq unknown, versus H,:oZ # r * , p a n d  qunknown,canbebasedonthestatistic N ( p , u 2 ) ,  --CO < p <-CO, 0 < <-CO. N(v,z’) ,  --CO < 77 < 00,0 < Z’ < 00. where x = T X i /  i 4  n and j=1 [40 marks] (a) Nyatakan Eema asasi Neyman-Pearson. [lO markah] (b) Huraikan konsep-konsep yang berkaitan di dalam teori pengujian hipotesis. [20 markah] (c) Katakan X , , X ,  sampel rawak saiz 2 daripada suatu taburan dengan j k .  k. + e ) x e  ,o I x I i,e > o 0 ,ditempatlain f ( x ;  9) - . . .81- 8 [MST 5611 Tunjukkan bahawa ujian yang paling berkuasa secara seragam untuk menguji H a  : 9 = 2 lawan Hi ;9 = 9, < 2 diberi oleh Apakah saiz ujian di atas? f30 markah] (d) Katakan XI, X, ,..., X, sampel rawak daripada taburan dengan$k.k.. Katakan Andaikan dua sampel tersebut tak bersandar. Tunjukkan bahawa ujian nisbah kebolehjadian untuk mengqji Ha : oZ = r 2 , p  dan q anu, lawan H, : Q z t ’, ,U dan 7 anu, boleh didasarkan pada statistik. N(F~cT~) , - co<,u<co ,  O<& <CO. ,Yz ,..., Ym sampel rawak daripada taburan dengan f k . k  N(7,rZ) ,  --CO < < 00,0 < 7’ <CO. C(yj -qz/ (m -1) j=1 F = T X i /  n dan j=1 di mana i=1 f40 markah] . . .9/- I 0 0 0 0 0 0 0 c 

MST 561 - Statistical Inference - Oktober 2004

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MST 561 - Statistical Inference - Oktober 2004

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