{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–8. Sign up to view the full content.

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ) ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 8

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

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online