Stat 543 Exam 2 Spring 2016

Transcription

1 Stat 543 Exam 2 Sprng 2016 I have nether gven nor receved unauthorzed assstance on ths exam. Name Sgned Date Name Prnted Ths Exam conssts of 11 questons. Do at least 10 of the 11 parts of the man exam. I wll score your best 10 answers at 10 ponts apece (makng 100 ponts possble). There s also on the last page of the Exam an "Extra Credt" queston that wll be scored out of 10 ponts. Any Extra Credt obtaned wll be recorded and used at the end of the course at Vardeman's dscreton n decdng borderlne grades. DO NOT spend tme on ths queston untl you are done wth the entrety of the regular exam. 1

2 1. Below are three pdfs for X, f x 1, f x 2, and f x 3. Use them n the rest of ths queston. x a) For whch are there non-randomzed most powerful sze tests of H 0 : 1 vs H 1: 2? b) Identfy a most powerful sze.15 test of H 0 : 1 vs H 1: 2. 2

3 c) Fnd a 0-1 loss Bayes test of H 0 : 1 vs H 1: 2 or 3 for a pror dstrbuton wth 1.4, 2.3, and 3.3. (Gve all 7 values of x g g g.) 2. In ths problem we'll use the Exp dstrbuton wth pdf f x exp xix 0 use wthout proof the facts that f X Exp and 0 t then P X t exp t,. You may f X ndependent of X Exp then Y mn X, X Exp 1 Exp In a so-called "competng rsks" context, an ndvdual or tem has a lfetme Z mn UV, U and V are postve tmes to falure/death from two dfferent causes. a) For 1,, n model U Exp 1 and V Exp2 what s observed are the d pars W Z, IZ U. (Note that I Z U observed s the value of U and the fact that U V w z,1 and z,0 w. where wth all U's and V 's ndependent. Suppose that =1 means that what s.) Gve lkelhood terms f, f w,1 1, 2: f w,0, : 1 2 w for observed 1 2 3

4 b) Sometmes, a cause of falure may not be recorded and thus only Z (and not w ) s known. Suppose that nformaton on 5 w1 3,1, w2 7,1, Z3 2, w4 3,0, and w 5 1,0. Suppose further that a Bayesan uses a pror for 1, 2 that s one of ndependence wth both 1 and 2 a pror Exp 1 dstrbuted. Carefully descrbe a Gbbs samplng algorthm for generatng trples * * 1, 2, w3,2 j j j * n ndvduals/tems s (terates for the 2 rates and the unobserved ndcator). If t s possble to name a dstrbuton from whch a gven update must be sampled, do so. At a mnmum, gve a form for each unvarate update dstrbuton up to a multplcatve constant. 4

5 c) Completely descrbe an EM algorthm that can be used to fnd an MLE of, based on the data used n part b). (It s not really necessary to resort to EM here, as the calculus problem s farly easy. But for purposes of the exam, wrte out the EM algorthm.) 1 2 5

6 3. Suppose that X1, X2,, Xn are d Ber p. Let S m m X X avalable for nference develops an estmator 1. A statstcan expectng to have only n 1 observatons S n 1 for p under SEL. (Ths estmator may well be a based estmator.) In fact, n observatons wll be avalable. Fnd another * estmator of p, say, that you are sure wll have smaller MSE p than the value of p 0,1. S n no matter what s S n 1 6

7 4. Suppose that X1, X2,, Xn are d wth margnal pdf f x x 1 I0 x 1. a) Fnd a lower bound for the varance of any unbased estmator X of sn vector of n observatons). (based on the b) Do you expect there to exst an unbased estmator of sn Explan! achevng your bound from a)? 7

8 5. Suppose that X1, X2,, Xn purposes) are d B, mp and (perhaps for "acceptance samplng" p P X p p s of nterest. Fnd an UMVUE for ths quantty and say why you know your estmator s UMVU. m (Hnt: You may fnd t useful to thnk of the each Ber p.) m X 's as Y, j for mn ndependent varables Y, j1 j

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10 6. Argue carefully that you could use d double exponental observatons (.e. ones wth margnal pdf 1 exp x on ) to generate a standard normal random varable va the rejecton algorthm, but that 2 you could NOT use d standard normal random varables to generate a double exponental random varable va the rejecton algorthm. 9

11 7. (EXTRA CREDIT ONLY) Consder the two margnal pdfs and d observatons 1, 2, 2 f x 0 I0 x1 and f x 1 I0 x 1 ln 2 X X from one of these dstrbutons (specfed by f H : 0 vs H : 1 for any sze 0,1 that there exsts a non-randomzed UMP test of 0 1 X,, 1 X n. Gve an explct large n approxmate form for such a test for.05. x x ). Argue carefully based on 10