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**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
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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
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