Monrad.410.36 - STAT H10 5: HONRAD E M “ya 3 i X”...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: STAT H10 5: HONRAD E M “ya 3 i X” Wfipwawfi Poisson M46441 9;}0‘ _ ) e < CO I TRL HLE E, 9 ZQYIG kk WKA Ha Qxé H:@:@O WSM HAé’gieD} 1Cor 5W SEQ/196:} (WM (95 > O, NOR Since. W W allot/m ‘ 0W}: MW’SMLM ) is NO UMP I? (m t ’ ‘~‘-' {+6.20%an «Hm. Olzel‘fkooai WHO 1164+ s‘zfighc‘; is K CD \/ ! /\ "- 2M : “Veéye‘wea _ He) (XBYGE'Y Assam/M Wfleméé <‘ % <16 Lg} 0<c1<c2 618/va ’lLl/l/LMfia/M far avg/CA /\(c:1) = /\[C;§ : A g /\é ha QM 0mg ‘er Yé C1 or Y9 CZ“ ($611 Ufldm ’Ha) Y is Poisson LLNHX WWW/x H900 M Mayan “Wot HO 1? /\ 4 A ” Wok is W SW ow “ Ragech H0 “H6 aw Veg or Y ‘2 ca” m si‘Wfimm {we/K 05' “ PeiYéC'r] + Pee”? 52] Er 90:2 M fl=5} {x =— Peewé Li] + Pep/2,17] = 0,020: + 0.027 : 09056 STAT mo / MATH «4w Exercise 3., 2.5 The joint pdf of X1,X2,...,Xn is 1 11/2 1 n 2 L(9;931,a:2,...,:cn): fl exp . 355 . . Let 0” represent a number gL—hw than 9’ , and let k denote a p081tive number. Let C be the set of points Where L(0/ 1x173327' ' ' 7x77.) L(9” 2 £171,182, . . . ,LUn) that is, the set of points Where all n/2 all _ 6/ n 2 (W) exp l_( 26’6" > 23% S k I I u ' I O or, equivalently, Slflg; e "' Q J n 20/6” n 61/ g 6,,w0, [510g ~logk] =0. 1 The ilarcslmlol 0 depends on 06 n and 6' buil- viot on 9",. 'n, The set C : {(131, 3:2, . . . ,rrn) : c} is then a best critical region for testing 1 the simple hypothesis H0 : t9 = 0’ against the simple hypothesis 6 r: 6”. It remains to determine 0, so that this critical region has the desired size a. If H0 is true, the TL random variable /6' has a chi—square distribution with n degrees of freedom. 1 , Since oz 2 P9, /l9’ S c/O’), c/B’ may be read from Table II in Appendix 1 TL B and c determined. Then G 2 {($1,3v2, . . . ,xn) : is a best critical 1 region of size a for teFting H0 : 0 = 6’ against the hypothesis 6 = 6”. Moreover, for each number (9” & than 0’, the foregoing argument holds. That is, C = n {(531, . . . ,xn) : S c} is a uniformly most powerful critical region of size a for 1 testing H0 : 0 = 0’ against H1 : 9 < 0’. If 131,1:2, . . . ,azn denote the experimental values of X1,X2, . . . ,Xn, then H0 : 6 = 6’ is rejected at the significance level a, 'n. and HA : 0 < 6’ is accepted if S 0; otherwise, H0 : 0 = 0’ is accepted. 1 Exercise, £2. 4. Exmple 8.2.2 sham «Wm/r iékjké‘ Ham - “a; (754’ Cf‘ilr‘wca rtgéon fir 'n ‘MKI._(L_$W (€— . h “(aegis H t 9 = ’ aaind- Hue Sim 1: Wowsk H‘,’ -. e -— a” his. {fine 5m?“ C ‘ §(x.,...,m3: 2c§ I Mm 'Hflt HM‘QLKOIOK C. depends on 06, W, 00M 9‘ . Since “MS” is a diqercnf region 1Com HM one in Exec—aka 8.2.5 J Men is no umformlg Mosvl- {some 46% {M 4646145 H0 : 9 = 9' againsf HA : 9 =1"- 6’, Exercigg g 2.3, Ar win as in EKM {6 (3.2.3 [4 H62 éem“ WHCA/(angion "QT fi'njeifv=9325 ' ajalhsl' e .S/MP/c 'Ag/DO'MCSIS Ht: :0 6” marcheugmresenk mtg mméer Hum ’25; :2 C -‘-‘- 5(x“...‘>(.): 3(— é CE 1 (Dem-e “UL “we;th (1 depends on oz MA n. 0-") = 942% = P<Xscte=zs> -—- fl—C—MZ‘ZSJ 0.610 = 7(23) = P(§Zscle=23) = @(J’Mfy—ifl) This 3mg as #u 2 eflmkons ——C———Mq25 = -I.Z&Z and m = L282 wex’uch imp‘g C. = 2% and n = 26.3 a: 26 Exercise 8.3.4 A -_- _CL 6.1 : .(21r)'"’2.o<et~ifixg-e’F/2l Lb?) (m)‘"‘« exp[ ~ 2 "sz (X3‘X = upf~n(3<‘-6')"/ZJ é 3» H: and onlg 16 (£3631 3 1% am aflg .4 (5229'! a \I-fiwx‘) The, [ila‘ilnood who {03+ 1%ch; H°= 6 =9! in \Cnmr 0? HA‘ 6 *9‘ i? and 0M3 {\C 19'?- 9" 2 Hm?) e {*kreskoM C. = V-E-Mx‘) is dckrmimd b w ("V-WC) = 06/2 . Exampk 8.2.3 Show} \‘aAL Hurt. is no unif-ormlg mod- {30mm MFx‘ov- Ho againslc HA. Exgrcise. (3.3.7 0 = {(61,62), —oo < 61 < oo, 0 < 62 < 00}, w = {(61,6'2), —00 < 61 < oo} L(Q) 2 L(01,62; $1, . .. 7£137?) : (27r02)_”/2ea:p{— Sign» — 01)2/(202)} Zn(LUD)::~§hfi2w6fl-—Z;5&E:EL 26 0=&%@Lfl;%giflea;iggm=X 0 : = fir; + 44“: 2‘22” => 0‘2 =3; — X)? Mm = L<é1,éz;xl,..,n> = = ln(L(w)) = ~~§ln(27r0,2) “ 2122—902 = W : 22fi=2;:i“61):>él I l i=1 532' = X Lo) = Lax, 0;; = (Ma-mem— grzlccg — 262/969} A = = (—L-J—-—EZ‘=H;:“X>2>n/2exp{—e+———Z'= if“? = (%)H/2exp _ 2 Where Y : Zfzficvi — X)2. From the plot of A vs Y we notice A S /\0 iff Y 3 cl or Y _>_ c2 Where cl, C2 are constants depending on /\o Chosen appro— priately. Exerc 1'52 (93.8 a) Under the general hypothesis (2, X1, ..,Xn ~ N(61, 03) Y1, .., Ym ~ N(62, 64) ——— L(61,62,(93,04;X1, . . . ,Xn,}/1,. . . ,Ym) Z . (Iii—9 2 _ mi 1 iyg 2 : (2W63)~n/2e$p{___t52(;93“1)}(27r64) m/2e$p{_ =52: 2) : Z L(617637$1,..,n)L(027647yli'Wm)‘ In the previous problem we saw that 1,091, 63, is maximized by A _ A _ A A we it $._ — 2 61: X, 63 = i r 1($1.402, ,L(61,03;m1,..,n) = (--———————2 2'11} ‘ X) W” n 2: Similarly for Y variables we get _ 62 '= Y, 04 = firm — W, ,L(62,64;y1,..,m> = (WW2 Hence : (w)—n/2(2wez.=ml(y;—Y)2)_m/2 Under CU X1,..,Xn, Y1, ..,Ym N N(61,63) n ._ 2 m ._ 2 Lad) =L(.91,62;X1,..,Xn,iq,..,Ym) =(2w03)—<“+m>/2exp{—M= (x‘ 6” +2: (7" a” } 203 A '1! Xi+ 7’1, K ‘ Zn 2m [1(a) : n21; [221:1 —' U22 + (yi —- u)2]}“(n+m)/2 A Z 2 {[ [2:35:33/n1::2(E%2ili;::ifl:gfl b) 9 is the same. Under w X1, ..,Xn ~ N(01,03), Y1, .., m N N(62,03) L(w) = (27r03)‘"/2exp{—W}(2w63)_m/2exp{—W} 6ll 2 X7 52 = Y» 533 = nim(§3?=1(wz-~ X)2 + 22-11(92' — 37?) 11(5)) 2 (gmé3)—(n+m/2) = [memmw ,\ z are; 2 (2:;(weir/ninfltzzl(yew/mm” [ __ [m+n 1 _ ]n 2[m+n _ 1 _ ]m/2 * 7” 1+Zi=1(yi“Y)2/Zg=1($i“X)2 m 2:1(xi-X)2/Z;=1(yi—Y)2+1 m n n m n 1 m 2 ___ . Zl :5 1+cn—1)/in—1)F-ll fll {rt W] / —functwn(F) 9‘ ...
View Full Document

This note was uploaded on 06/11/2010 for the course STAT 410 taught by Professor Alexeistepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

Page1 / 7

Monrad.410.36 - STAT H10 5: HONRAD E M “ya 3 i X”...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online