Stat 543 Exam 2 Sprng 206 I have nether gven nor receved unauthorzed assstance on ths exam. Name Sgned Date Name Prnted Ths Exam conssts of questons. Do at least 0 of the parts of the man exam. I wll score your best 0 answers at 0 ponts apece (makng 00 ponts possble). There s also on the last page of the Exam an "Extra Credt" queston that wll be scored out of 0 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.
. Below are three pdfs for X, f ( x ), f ( x 2 ), and f ( x 3). Use them n the rest of ths queston. x 2 3 4 5 6 7 3.20.05.5.5.0.05.30 θ 2.0.05.25.05.20.5.20.25.5.05.25.5.05.0 a) For whch α are there non-randomzed most powerful sze α tests of H 0 : θ = vs H : θ = 2? b) Identfy a most powerful sze α =.5 test of H 0 : θ = vs H : θ = 2. 2
c) Fnd a 0- loss Bayes test of H 0 : θ = vs H : θ = 2 or 3 for a pror dstrbuton wth () =.4, ( 2 ) =.3, and ( 3 ) =.3. (Gve all 7 values of ( x) g g g φ.) 2. In ths problem we'll use the Exp( λ ) dstrbuton wth pdf f ( x λ) λexp( λx) I[ x 0] use wthout proof the facts that f X Exp( λ ) and 0 t > then P[ X t] exp( λt) > =, = >. You may f X ( λ ) ndependent of X Exp( λ ) then Y mn ( X, X ) Exp( λ + λ ) Exp 2 2 2 2 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 =,, n model U Exp ( λ) and V Exp( λ2) what s observed are the d pars W = ( Z, I[ Z = U] ). (Note that I [ Z U] observed s the value of U and the fact that U V w = ( z,) and = ( z,0) w. = where wth all U's and V 's ndependent. Suppose that = = means that what s <.) Gve lkelhood terms f ( λ, λ ) f (( z, ) λ, λ 2) : ((,0 ), 2) f z λ λ : w for observed 2 3
b) Sometmes, a cause of falure may not be recorded and thus only Z (and not w ) s known. Suppose that nformaton on n = 5 ndvduals/tems s = ( 3, ), 2 = ( 7, ), Z3 = 2, 4 = ( 3,0 ), and 5 = (,0 ) Suppose further that a Bayesan uses a pror for (, ) w w w w. λ λ 2 that s one of ndependence wth both a pror Exp() dstrbuted. Carefully descrbe a Gbbs samplng algorthm for generatng trples * * * (( ),( 2),( w j j 3,2) j ) λ and λ 2 λ λ (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
c) Completely descrbe an EM algorthm that can be used to fnd an MLE of (, ) λ λ 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.) 2 5
3. Suppose that X, X2,, Xn are d ( ) Ber p. Let X avalable for nference develops an estmator ( ) S m m = X. A statstcan expectng to have only n observatons δ S n 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 δ ( ), that you are sure wll have smaller MSE p than ( ) the value of p ( 0,). S n = δ no matter what s S n 6
4. Suppose that X, X2,, Xn α are d wth margnal pdf f ( x α) αx I[ 0 x ] = < <. a) Fnd a lower bound for the varance of any unbased estmator δ ( X ) of γ ( α) sn vector of n observatons). = α (based on the b) Do you expect there to exst an unbased estmator of ( ) sn Explan! γ α = α achevng your bound from a)? 7
5. Suppose that X, X2,, Xn are d B (, ) mp and (perhaps for "acceptance samplng" purposes) ( p) P [ X 0] ( p) γ = > = p s of nterest. Fnd a UMVUE for ths quantty and say why you know your estmator s UMVU. m (Hnt: You may fnd t useful to thnk of the Ber ( p ).) m X 's as Y, j for mn ndependent varables Y, j each j= 8
6. Argue carefully that you could use d double exponental observatons (.e. ones wth margnal pdf exp ( x ) on R ) 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
7. (EXTRA CREDIT ONLY) Consder the two margnal pdfs and d observatons, 2, 2 f ( x 0) = I[ 0 < x< ] and f ( x ) = I 0 < x< ln 2 x ( ) [ ] X X from one of these dstrbutons (specfed by f ( ) that there exsts a non-randomzed MP test of H 0 : θ 0 vs H : θ = = for any sze ( 0,) X,, X n. Gve an explct large n approxmate form for such a test for α =.05. x θ ). Argue carefully α based on 0