409Hw06ans - STAT 409 Fall 2009 Homework #6 (due Friday,...

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Unformatted text preview: STAT 409 Fall 2009 Homework #6 (due Friday, October 9, by 4:00 p.m.) 1. Let a > 0, 9 > O and let X1,X2, ,Xn be arandom sample ofsize n from a distribution with probability density function fX(x)=fX(X29)=%'Xa_1, 0<x<9. Suppose a is known. a) Find the sufficient statistic Y = u (X 1 , X2 , , X n) for 9. N b) Find the method of moments estimator 0 of 9. ‘03 ? & Xavl A, _ a9 , M'— x .. ~.—————— EBA X 9“- CL-H O .—. .0195) rv- CL+\ R X = ’7‘ '7') 9 ~ “*X. a—H 6L b) Leta > 0, 6 > 0 and let X1, X2, , Xn be arandom sample ofsize I’l froma distribution with probability density fimction fX(x): fx(x;9)= a _ ——-xa 1, O<x<6. 6a Suppose a is known. A Find the maximum likelihood estimator 9 of 9. A Is 9 a consistent estimator of 9 ? Justify your answer. Hint: Let8>0. Find P(|é—e|za)=P(ése—g)+p(éze+g). LL63 —. the :83 O4x\49 0V vx a“! mzax; >0 «pa- n K\ {Q {9" {7,4 9 éj mm a a MM X2 4‘3 : OéKHC‘B 0 UL»). 3. Let a > 0, G > 0 and let X1, X2, , Xn be arandom sample ofsize n from a distribution with probability density function fX(x)=fX(x;9)=%c;'xa—l, 0<x<9. Suppose a is known. a) LetO<oc<1. FindcsuchthatP(69<maxXi<9)=1—oc. (C would depend on n, a, and 0t.) b) Suggest a ( 1 ~ CL) 100 % confidence interval for 9 based on max Xi. QJ P<Q94W¥i L9\ :. i'FWy:(C‘?-33 ’ ” r- _ 92.“ —. [. L?x(ce3j _ i (9} : k’C’kC/‘x a; \«cm‘:l—0L. :? («=4 b) Let a>0, 9 >0 and let X1, X2, ,Xn bearandom sample ofsize n froma distribution with probability density function fx(x)=fx(x;9)=ia'xa_l, 0<x<6. 9 Suppose G is known. Find the sufficient statistic Y = 2,1(X1 , X2, ,Xn) for a. n X _ What is the probability distribution of — Z 1n ?1 ? i=1 0% (1 7K; 2 wttwq I :r‘ e an“ {74 I h 9% Rubs—mum. W, ‘t = U x{ ~‘ “A :s satme £9-(6L_ OR £(X'lo.\ : 9/,xf<_(q-\\Qkx V+QM0u —cLQ/n€~>"], Khdt- wa. v> :Z:K(¥;‘):iQMX‘ is ‘L‘ w: who/4%, WW“: PC‘LQQ 9Q = ? (X2 966$ 1“ \-— Y’xtez‘“) : \ 1—r 3:1“ ck : \vefoval ‘fivck “Rik, .ch am :cA exewwi w» W :~> {kc Mg (manna (law, 9:153 NbRX-m} M 5. Let a > 0, 9 > O and let X1,X2, ,Xn be arandom sample ofsize n from a distribution with probability density function fX(x)=fX(x§9):“qz{'xa_la 0<X<9- 9 Suppose 9 is known. a) Find the maximum likelihood estimator & of a. b) Find the method of moments estimator c7 of a. A V‘ 6; AA — \ CL“ V‘ 0~ ~ ’\ a.» T. m 1X; :— mg.) i :i 9 9 x f .4 QAA L(6‘\\: nQ/wa ~chL Q/hQ + (cg—43 EAQMXIL V\ iL(a\:z\f~\/\Q«9+ZQHXT A“ 4’1— {'4 . A V\ :L L(&\ 1 O 2—,”) av: M : R 6‘ All/m? -— EKG/“Xi ,: m a 6x! 0&9 CK . ' :2" .. ,. f x CL ; LOO ” g X 8“ X a-H o X = £32 :7 0x = ’ 5L+l 9..)C b) 53 Let a>0, G>0 and let X1,X2, ...,Xn bearandomsampleofsize I’l froma distribution With probability density function fx(x)=fx(x;9)=ia'xa"l, 0<x<6. 6 Suppose 6 is known. Consider £1 Is 51 a consistent estimator of a? Justify your answer. ’1 3 32x1 A Consider a = 121 n . Is a aconsistent estimator of a? 3 3 "e _ ZZIXZ' Justifiz your answer. I: a :. I WM. X : h <1; X} r 8":— 327- M ') 9 l -"‘ 1 ~ L 6x Cs" «9 o.“ WLLM) XL MEKX3=§Xm-ax bkx: . o 9 CK‘R'L r2' '2. i X C$ Wk‘KMkWS‘ agk Kg elflfit 6K4”). me“ A ~— '? - 7-— 2. . :7 *3: 30(1) “hogiae = ““L :C’L‘ (if), 91' Ciel one"). /\ A {‘5 cs W$zsm 52"“ PNch (A A i «'3 ._.——‘-— n r-S a ‘1 5(1) ’ , ' ‘.\ a 2x 9 ‘ S63 0. xa‘AM' 0‘9,3 "-3 ' "> = X ‘ c v ’ . (‘AQ \ULLIV I x ~39 ELK 3 o as (US Q93 6‘ (7:3: 3"" ;» Wswm wt , A '5 Q‘VB e I’X’, P awed" 3 6L9": A ,«—-— ' w : dink") _ Bax . a») a1?“ 33MELa-x—g W~——#a engu—w wk A an”: a {5 6‘» W‘MfiM igivmw- ew- C)\ - 7. Let X1, X2 , , X n be a random sample from the distribution with probability density function 3 2 9 +36 +26 x9_1( 2 1—x)2 0<x<1 6>0. f (x;9)= a) Find the sufficient statistic Y = u (X 1 , X2 , , X n) for 9. N b) Find the method of moments estimator of 9, 9 . m - _ , V‘ K S”\ h as m {1(x;‘,9\ {W} Q: at many. {fix ’2' \=\ {2" ‘ . O ‘k : S 9(9+\\t9+2—\ (X9-1x9+‘ +XGJH)‘ 0*“ Q, 1. 041 X139“- ,Xm wrt “A B{m(4:el(;:'s\_ :7 600: (it‘s: i: “)2: g 7'7 %: 3:2 8. Let X1, X2 , , X n be a random sample of size n from the distribution with probability density function fx(x)= fx(x;6)= <e~1>2-m—;‘, x>1, e>1. x ’1 Recall (Homework 3) that ZlnXl- has Gamma( on = 2 72, “usual 9” = ————1—1— ) distribution. i=1 _ ( “Usual” Gamma( 0L, 9) distribution: = 1 x (1—1 e'x/e , O S x < 00. ) F(a)6 ‘1 a) b) 71 Suggest a (1 — on) 100 % confidence interval for 9 based on ZlnX l- . i=1 Suppose n = 5, and Construct a 95% confidence interval for 9. 0% vie/“x; we; GucMMaH 27m, “meant e“ ._ 0U$HWM s‘”\ 2 22. flax: {Li H \U‘avudl G“ :7 = 2(9 —0 bx; Mg 7C1 (2:; :th bath-mom \‘3{ La}: C:%T,in(Ltv-\\ ) :7 ’3. g < 1(9*\3::‘QMX; 4a\ __O C. . ‘ _ (51 *Hk‘kzéafix; éegh—ffl: TH .— _ H ' Lin *3) (\+ 76\ ELL‘\ \+ 7‘9:( \ B :s a (t—umoO‘Z, " . ) libs lELX; . \“ I=\ :«hvk—Q b3 “13);:09‘5‘ C277jqfitm3;q,sci\, Azy'fiwgcao) -_ awn) :1 Panzssasz . 615m 79*“? W ( + W —v 38‘3 L\.\% . (‘ $95133 ) i ’L«§-°>73‘> i ) ...
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This note was uploaded on 10/12/2011 for the course STATISTICS stat 410 taught by Professor Stepanov during the Spring '11 term at University of Illinois, Urbana Champaign.

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409Hw06ans - STAT 409 Fall 2009 Homework #6 (due Friday,...

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