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**Unformatted text preview: **M3/M4 Imperial College
London BSc4 MSci and MSc EXAMENATIONS (MATHEMATICS)
May — June 2015 This paper is also taken for the relevant examination for the Associateship of the
Royal College of Science. Statistical Theory | Date: Monday—1i May 2015. Time: 2.0% — 4.00pm. Time allowed: 2 hours. This paper has FOUR questions. Candidates should use ONE main answer book. Supplementary books may only be used after the main book is full. Statistical tables are provided on pages 5 81¢ 6. o DO NOT OPEN THlS PAPER UNTIL THE INVIGELATOR TELLS YOU TO. 0 Afiix one of the labels provided to each answer book that you use, but DO NOT USE THE
LABEL WITH YOUR NAME ON IT. a 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. i—Raw mark Mate 12 13 14 T13 16 ii? l 18 197 207
lExtracrediti 0 % iiié 2 '2%l_3 3&1 41 a Each question carries equal weight. :- Calculators may not be used. (9 2015 imperial College London M3/M4 51 Page 1 of 6 1. (a) Let X denote the observed data, whose distribution depends on an unknown parameter 9 E 9. Let T = 39:") be some statistic. Write down the definition for each of the
following concepts: [i] T is sufficient.
(ii) T is minimal sufficient. (iii) T is complete. (b) Let X1, . . . ,Xn be i.i.d. Exponentiaﬁd) random variables. (E) Show that X is the Crame’r-Rao Unbiased estimator of some function p09} of 6, and
write down its variance. (ii) Find a variance stabilising transformation 9 such that verge-it) — go» 1: Mo, 1)- (iii) Show that for such Exponential(6) observations, the Gamma(o.;3) prior is a
conjugate Bayesian prior. (iv) Let the prior distribution of 6 be Gammafoc, ,8]. Compute the posterior rnode. ix.) (a) Let 9 E 9 be an unknown parameter and let X denote the observed data. Consider the
null hypothesis Hg : 8 E 60 and alternative hypothesis H; : 6 E (-31 = E) \ 60. (i) Give the definitions for the size a and the power function it? of a hypothesis test with critical region H. You may use the notation P3_t0 denote the dependence of
the probability measure on 6. (ii) Give the definition of an unbiased test. (iii) Explain how one .can construct a 100(1 - ci-)% confidence intervai for 6' by first
considering. for various 80. size o: tests of Hg : 9 = 9;]- v.s. H1 1 g )1: 90- (b) Let Y}, . . . , 71”,.l be independent with K- ~ .Mifdzrﬁg). where 03 and the 23.; are known
constants. (E) Find the Cramér~Rao unbiased estimator for 49, and write down the corresponding
Cramér~Rao lower bound. ' (ii) Find an unbiased estimator of 6 which is a function of 37' = i 23:] K. What is the
efficiency of this estimator? (c) Suppose that we observe both X N Geometricfl — 6) and Y m Poissonlﬁ). X and Y
are independent. 49 6 (0,1) is an unknown parameter. (i) Write down a minimal sufficient statistic t(x,y) for 6‘.
(ii) Show that the likelihood satisfies the monotone likelihood ratio criterion. (iii) Does a similar uniformly most powerful randomised test of size a = 0.01 for testing
H0 : d = 0.5 against H; : 6' < 0.5 exist? Write one sentence justifying your answer. M351/M4Sl Statistical Theory | (2015) Page 2 of 6 3. Let X1, . . . ,Xn be i.i.d. samples with PMF fA-(x) = 9(1— SHEILQO. and unknown parameter
6 E (0,1). This distribution is an alternative version of the geometric distribution with range {0,12,...}. (3) Justify without proof why S = 2;} X,- is a complete sufficient statistic for 9. Find an unbiased esrimator for 62 in the case where n = 1. by comparing coefficients of
(1— 9V in a suitable expansion. Compute the total score function LEW} and the total Fisher information i.(6).
Explain why there is no Crame’r-Rao unbiased estimator of 8. 133:0 is an unbiased estimator for :9. Assuming 'n. > 1. obtain an improved estimator by
applying the Rae-Blackwell procedure using the sufficient statistic 5'. [Hints S foiiows a Negative-Binomiai distribution with range {0, 1, 2, . . } because it is a
sum ofn Geometric random variabies with ranges {0, l, 2, . . .}.j is the improved estimator obtained in (e) the minimum—variance unbiased estimator for
19? Justify your answer. M351/‘M451 Statisticai Theory | (2015) Page 3 of6 4. Let .X'1,...,X.n be i.i.d. samples from a Uniformiﬁﬂx) distribution‘ and let Y3,...,Yn
be independent i.i.d. samples from a Uniformioﬂy) distribution. We are interested in
the hypotheses Hg : ex = 8y and H1 : ﬁx ;’ 9y. Let Xin) = max(X1‘...,Xn), let
Ym = mEDL(Y1, . . . 3 3;.) and let T : maxLX'l‘ . . . ,Xn, Y1, . . ., n) 2 maxiXm, Yw). (a) Show that (Xw, Yinii is a sufficient statistic for (9X1 By]. (b) is the hypothesis H0 simple or composite? is the hypothesis H1 simple or composite?
You are not required to justify your answers. sapBX=9y6{0.m)L[5X‘9Y) E5
the (generalised) likelihood ratio. Assuming H0 is true‘ show that A is ancillary for 6=6X =Eiy. (c) (i) LetA=A(X1,...,XmY1,...,Yn)=210g(,\),where)i=W' (ii) Assuming H0 is true, show that the distribution ofA is XE. You may use the following
facts without proof: 1. If A ~ Xi: is independent of B m xi then A + B m Xgch.
2. if A is independent of B N xi and A+ B m X1271“: then A m X;-
. If Z w Betailﬁ) then —26iog(2) «2 X3- 3
4. 3%?) m Retail, to), independently of $31 m Beta(l,n).
5 _ if 8 = 8X 2 By then T is a complete sufficient statistic for 6 and g m
Beta(1,2n). [Hintz First show that T is independent oi‘AJ (iii) Compute the critical region for a likelihood ratio test of size or of Ho v.5. H1. [Hint The )6 distribution, the Exponential (é) distribution and the Gamma. (1., %)
distribution are aii identicaij (d) You may assume without loss of generality that n is large. Comment on how the
distribution in (c)(ii) relates to Wilks' Theorem. Explain why this test violates the
regularity conditions for Wilks’ Theorem. [Hint The reguiarin/ conditions for Wiiks'
Theorem are the same as those given for the asymptotic normaiity of maximum iikeiihood estimators. The reguiarity condition which is vioiated here is aiso one of the conditions
needed to prove the Crame'r—Rao iower bound] M3515’M4Sl Statistical Theory l (2015) Page 4 of 6 . N. . b . .
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- Null hypothesis, Statistical hypothesis testing, Statistical theory, unbiased estimator, Wilks