M3S1 2016.pdf - Imperial College tandem M381 BSc MSci and...

This preview shows 1 out of 12 pages.

Image of page 1

Subscribe to view the full document.

Image of page 2
Image of page 3

Subscribe to view the full document.

Image of page 4
Image of page 5

Subscribe to view the full document.

Image of page 6
Image of page 7

Subscribe to view the full document.

Image of page 8
Image of page 9

Subscribe to view the full document.

Image of page 10
Image of page 11

Subscribe to view the full document.

Image of page 12
You've reached the end of this preview.

Unformatted text preview: Imperial College tandem M381 BSc, MSci and MSc EXAMINATIONS (MATHEMATICS) May—June 2016 This paper is also taken for the relevant examination for the Associateship of the Royal College of Science Statistical Theory 1 Date: Tuesday 10th May 2016 Time: 09.30 — 11.30 Time Allowed: 2 Hours This paper has Four Questions. Candidates should use ONE main answer book. Supplementary books may only be used after the relevant main book(s) are full. Statistical tables are provided. - DO NOT OPEN THIS PAPER UNTIL THE lNVlGILATOR TELLS YOU TO. . Affix one of the labels provided to each answer book that you use, but DO NOT USE THE LABEL WiTH YOUR NAME ON lT. - Credit will be given for all questions attempted, but extra credit will be given for complete or nearly complete answers to each question as per the table below. Raw Mark 1 Up to 12 ’13 14 15 '16 17 18 19 I 20 Extra Credit 0 1/2 ’i 'i 1/5 2 2 1/2 3 3 1/5. 4 . Each question carries equal weight. ... Calculators may not be used. © 2016 Imperial College London Page 1 of 3 l. (a) State the Neyman Factorization Criterion and prove it for the case of discrete distributions. (b) Suppose that X1,... ,nX 1:3 Bernoulli (,6) where [l < 9 < 1 (i) Show that T: Z X- is a sufficient statistic for 6. is T complete? Why? Also argue whether or not T Is minimal sufficient for 8. (ii) Find the UMVUE of 9(1 — 6'). [Hint I(X1 = 0, X2 = 1) is an unbiased- estimator of6(1 - 9).] (iii) Compute the CraménRao lower bound for the variance of unbiased estimators of 6(1 — 9). Does the UMVUE of 6(1 — (3) obtained in (ii) attain this lower bound? Why or why not? 2. Let (X1.Y1),...,(XmYfl) be independent pairs of Normal random variables where X,- and Y!- are independent NW“ 02) random variables. (3) Find the MLEs of #1 ..... on and :72. (b) Now, suppose we observe only Z1,...,Z"n where Zi = X1- — Y}. (i) Find the MLE of 02 based on ZL...,Z,1 and discuss whether or not it is consistent. (ii) Obtain a method of moments (MM) estimator of (I2 based on 231,. WZ (iii) Consider testing Hg : 02 g org versus H1 : 02 > 00. Find the UMP test at level at based on Z1,... Z (iv) is the UMP levei cc test obtained in (iii) unbiased? Justify your answer. M351/M4Sl Statistical Theory 1 (2016) Page 2 3. Let )i'1,....Xn be i.i.d. random variables from the deiayed exponential distribution having the probability density function fem) = 9e“6($—2), a: > 2 where 6 is unknown. Suppose that the prior distribution for 6 is EXPOHEDtifl-MA) where A is a known positive constant. (3) Obtain the posterior distribution of 6'. (b) ls the prior here a conjugate prior? Justify your answer. (c) Find the Bayesian point estimator of 6' under the squared error loss function. (d) Verify whether or not the Bayes estimator obtained in (c) is admissible. 4. Suppose that X;, ..., Xm i323 Exponentialwl) and Y1, ..., Yn iii-ii Exponentialwg), and assume the Xi and the K- are independent. Consider testing Hg : 6'1 x 92 versus H; : 81 # 62. (a) Show that the likelihood ratio test statistic is as follows A(:.c)~m+n?_m”+mjfl ,y— m+n m+nX m+n m+n37 ' (b) Obtain a test at level 0: using the test statistic T); [Hint- Use the fact that if x? and X3 are two independent chiwsquared random variables 2 v with degrees of freedom tail and 1,12 respectively, then m m P (U1,U2). ] xifvz (c) Obtain the likelihood ratio test using the asymptotic distribution of ~210g()i(3:, y)) under H0. (d) Construct a confidence interval for 91 with confidence coefficient 1 — cr. 52 M351/M4Sl Statistical Theory l (2016) Page 3 b b E ”5.5% u E 35> E rain u E ém zaxézu n 3;: Amway é n 3E m A: m L 5‘ u 3:: 82m Kb + 1 H # nosmfiaommflafl EddamaZOHHq‘DOQ 2: 93 a .3: ale—Ian... \H GU: ZOHBOZD ra dggfiw 23 2&3. 69.5 $3 mnosafiémfi mDODZHEZOU Sh fl 4 a I a u I _ m a a . cm all QT]: a I a: as: :30 I ELEV : a w 3% w a #132 s giro; Nu m T: I . _ . I o. “um E 1.3,: m... Tube I Sam H I. H 3 a w a ._.N w a. .H H +: .5 E $3325:me v.2 be | Sun. I w «a m . . . . E a I g m 56 I s IH 3-53 I a a a w m A m z. EUEESQU .H E I 3 i 93 4 4 «h e E w 4 #133 Egfifi 3. T 1| L3 + a I a a I a? a: 3.3 I :3 @V 2 .3 w 3m w a it... 4 .3 $.::3.§Em “Na + a I H a I Q q 33 I :a c .a w a 2 .8 $5322.»: 2: at s. m . . C . .. “82 E 5% E 5. Eu 2 “mm? mmmemgfiém _ mwzé mZOHBDEMmBmwfl mahflfinumnn— L.“ w n5 2 ,8 3.333% E w a s E E £3245 E w a g Efiaam 3 H h. ,0 H a 313: c.3333 Auxuunuiiu +E w u .2 w 1 mm HnbixdEEZ 3 H 3. EEE Haw—53 E w u .a E EaEEEE 3., H m 3on 6.3.3433 “ham La w m :0 Ln $.6wafizEaU 2 H 4 ESE fiEccafiw W14 E w 4 +2 2:3.szaam 3 H Q4... H a. ESE Emwsfimu AH... I nu w n . II: N. V U 6 6 EE .3 37%. L643 8+3 fl g m a V a w a a 52 5. x 32 E $.52 E 5 moo r3; .mzéfi _ mZOHHDfi—fifimwfl mDODZHHZOU Imperial College London M431 BSc, MSci and MSc EXAMINATIONS (MATHEMATiCS) May—June 2016 This paper is also taken for the relevant examination for the Associateship of the Royal College of Science Statistical Theory 1 Date: Tuesday 10th May 2016 Time: 09.30 —- 12.00 Time Allowed: 2 Hours 30 Mins This paper has Five Questions. Candidates should use ONE main answer book. Supplementary books may only be used after the relevant main book(s) are full. Statistical tables are provided. 0 DO NOT OPEN THIS PAPER UNTIL THE INVIGlLATOR TELLS YOU TO. . Affix one of the labels provided to each answer book that you use, but DO NOT USE THE LABEL WITH YOUR NAME ON IT. . Credit will be given for all questions attempted, but extra credit will be given for complete or nearly complete answers to each question as per the table below. Uitoi2 Extra Credit “ a Each question carries equai weight. . Calculators may not be used. © 2016 Imperial College London Page 1 of 4 l. (a) State the Neyman Factorization Criterion and prove it for the case of discrete distributions. (b) Suppose that X.1._ ..., Xni'ig' Ber11oulii(9). where 0 < 6 < 1. TL (i) Show that T = Z X:- is a sufficient statistic for 6'. ls T complete? Why? 11—] Also, argue whether or not T is minimal sufficient for 9. (ii) Find the UMVUE of M1 — 6). [Hint I(X1 = 0, X2 = 1) is an unbiased estimator of6(1 — 6).] (iii) Compute the Cramér—Rao lower bound for the variance of unbiased estimators of 6(1 — 9). Does the UMVUE of 6(1 —- 6) obtained in (ii) attain this iower bound? Why or why not? 2. Let (X1,Y1) ..... (XmYn) be independent pairs of Normai random variables where X1; and Y} are independent Ngur, 02) random variables. (a) Find the MLES of #1,...,on and 02. (b) Now, suppose we observe only 21,...,Zn where 23- = X:- — Y}. (i) Find the MLE of 02 based on Z1 ..... Z71 and discuss whether or not it is consistent. (ii) Obtain a method of moments (MM) estimator of 02 based on Z1....,Zn.. (iii) Consider testing H0 : (72 g 03 versus H1 : o2 > org. Find the UMP test at level {2 based on Z}....,Zn. (iv) Is the UMP level 0: test obtained in (iii) unbiased? Justify your answer. M351/M451 Statistical Theory l (2016) Page 2 3. Let X1.....Xn be i.i.d. random variables From the delayed exponentia! distribution having the probabiiity density function fair) = file-“HI I > 2, where 6' is unknown. Suppose that the prior distribution for 6 is ExponentiaHA) where A is a known positive constant. (a) Obtain the posterior distribution of 8. (b) is the prior here a conjugate prior? Justify your answer. (c) Find the Bayesian point estimator of 6 under the squared error loss function. (d) Verify whether or not the Bayes estimator obtained in (c) is admissible. 4. Suppose that X1, ...,Xmiri§51'Exponentia.1(61) and Y1, ..., Yni'rii-‘riExponentiaIwzj, and assume the X1- and the Y,- are independent. Consider testing Hg : 61 : 62 versus H1 : 61 75 62. (a) Show that the likelihood ratio test statistic is as follows _ _m _ -7; M33 )_ m + n. Y T1 + m _.X: ,y '— m+n m—l—nX m-E—n m+n37 . (b) Obtain a test at level an using the test statistic “2;". [Hint Use the fact that if x? and 35 are two independent chi-squared random variabies with degrees of freedom U1 and o; respectiveiy, then gig; m F(o1,o2).] 2 (c) Obtain the likeiihood ratio test using the asymptotic distribution of —21og()\(2:, 3,0) under Hg. (d) Construct a confidence interval for g: with confidence coefficient 1 — 0:. M3Sl/M4SI Statistical Theory 1 {2016) Page 3 Mastery Question: 5. Let Xlr...,Xn be i.i.d. Cauchy random variabies with density function 1 W = W IER, and suppose outcomes m]....,rn are observed. (a) (b) (C) (d) (8) Write down the likelihood equation for estimating 6' and discuss whether it has a unique solution for the given sample I1....,33n. Given an estimate 5““) for 8 at iteration k, obtain a new estimate él'k'l'll using the Newton-Raphson method. Show that a new estimate 30¢“) using the Fisher scoring algorithm is n 332' _ fill“) gum) = gas} + E ____cfi_2. TL i=1 1 ‘l‘ (in; — film) . _ 3 I [Hint face @631: E] ls the sample mean 3‘: an appropriate initial value for the Newton-Raphson and the Fisher scoring methods here? If not, suggest a good starting point. Briefly explain your thinking. Which method has a faster convergence: the Newton-Raphson method or the Fisher scoring algorithm? Why? M3Sl/M4Sl Statistical Theory 1 (2016) Page 4 b .__u b E 53% u E :5» E émii u E 3Q saga“. n 3:: A: r L 2,... n 3...... m A: I L .é n 3.: 85% Rb +1 H > ”Baggage: ma<DmRZOHH¢OOQ 2: «:8 D . Hmc Hlmarafl \ H @6er ZOHHUZDH 46424.0 ME..— uamuu A53 83 mnosnnmsmzu m3 ODZHHZOD 5m 2.332%: fi....n.1$ s Q .3 w m .+N w z A: J + 2.5 megawawfiagmmuz sire; «a mm a | C m HS | E | H halanm | 3 9. .8 w n. Amuutuoqtnub “ . .8. w . n :M I 31. 95 4 4 a. u E m 4 A... m H a. 3523.... I|H|F||I| .9. true .T m | 1; be I Cm: a: slag 1 Sue DEV S .3 U Q .+N w .2 Thr: J. .Dw a rtuudwszcfimm 3 .8 w a 2 6.“ ASSEQEUQ wn ZOHHUZDE _ mm<§~ WMMHHQEé/wnm :— EOE/$.— $195: magma mgmoma _ H + +5 + .d + 5 : fl . . _ _ A a nmfla V a a 2%? :Taefiaa: E m n a : a a igum a A ca 2 A a a mlb |_O Hlfiv HIT 6?“.wa . . A Wm H lmll ab a av IH ~+ ah. I +m= w 6 a +2 cc 339.33. N . t a 1 I” + a g L m I a. a a A a a a 3 A a a c E w a m 33.53% a “Elfin; 2 H b .c N 1 .365 23:33 Pam Etnifine ab 3. .Vlw 98 N H +E w b in w a ”a 55.1335ch 7 3 H mm 3on Efiusfifl exam“ .I aim i \ .. . . _ _ E: + a u L Em + a L 3: + 3 L EL. M saufqa“? é w u a +2 E 3:33; C H a ESE ESE—3mg .I d . . Jamv mom. m HQIUwIGBE Jrfi w m. :0 +§ HQ.GHBEEEMU 3 n. 4 EEE @598ng mlwer WW. 4."; 21¢ 572 LE w < +w= QVHSEEEAHQ C. H .025 H d 355 Emwngfl Coin: 2 a a..n “Tn , . Malia L3 8. + 3 fl I H _ a w m V a. E S 3 EEEES .42 2,. i x. mm: -. E 53 E fim moo mam .Efimfi mZOHHDdeBED mDODZ—BZOU ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern