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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT0604 STATISTICAL MODELLING (I)
STAT0605 MATHEMATICAL STATISTICS December 18, 2000 Time: 9:30 a.m.  11:30 a.m. Candidates taking examination that permit the use of calculators may use any cal
culator which fulﬁls the following criteria: (a) it should be selfcontained, silent,
batteryoperated and pocket—sized and (b) it should have numeraldisplay facilities
only and should be used only for the purpose of calculation. It is the candidate ’3 responsibility to ensure that the calculator operates satisfactorily
and the candidate must record the name and type of the calculator on the front page
of the examination scripts. Lists of permitted/prohibited calculators will not be made
available to candidates for reference, and the onus will be on the candidate to ensure
that the calculator used will not be in violation of the criteria listed above. Answer Question 1 in Section A AND any two questions in Section B.
Marks are shown in square brackets. Section A 1. A dataset consists of 20 observations as follows: 1.388 3.312 0.745 0.153 0.177 0.061 6.108 1.295 0.212 2.851
2.029 4.459 4.949 2.607 2.187 5.871 1.811 1.543 0.853 1.485 The average of the 20 observations has been calculated to be :7: = 2.2048. Assume that the above data are 20 independent realisations of a random vari
able X with the density function f(x[9)=9"ln(0), x>0, 0> 1.
(a) Write down a likelihood function of 6 based on the observed data.
[4 marks] (b) Show that the maximum likelihood estimate of 0 is 5 = 81/5. Calculate
6. '
[4 marks] S&AS: STAT0604 STAT. MODELLING / STAT0605 Math. Stat. 2 (c) Show that the generalized likelihood ratio test statistic for testing
Ho : 6 = 60 against H1 : 0 unrestricted,
for some ﬁxed 60 > 1, is
A(60) = 40 {i‘ln 60 —1n(ln00) — lni  l}. [6 marks] (d) Show that MOO) monotonically decreases as 60 increases from 1 to 9, and
monotonically increases as 00 increases from 6 to 00. [8 marks] Describe graphically how you will obtain a 95% conﬁdence interval for 0
based on the generalized likelihood ratio test statistic. [ You are not required to give numerical answers here. ] (e) It is suspected that the assumption of the density function f(x  9) is
incorrect. To check this assumption, we summarize the observed data by
a contingency table of the counts of observations in four categories: Category OSX$11<XS2 2<X33 X>3 Count 6 5 4 5
The null hypothesis is that the observed data are drawn independently
from f (a: l 6). (i) Find the probabilities, in terms of (9, associated with the four cate gories, under the null hypothesis.
[6 marks] (ii) Using (b) and (e)(i), calculate the estimated counts in the four cat egories under the null hypothesis.
[4 marks] (iii) Test at the 5% signiﬁcance level for the null hypothesis based on the Pearson chisquared test and the categorized count data.
[8 marks] S&AS: STAT0604 STAT. MODELLING / STAT0605 Math. Stat. 3 Section B 2. Kids’ Kingdom, a popular amusement park, commissioned two consultants to
investigate visitors’ opinions. The consultants distributed questionnaires to
departing visitors separately, asking them questions as given below: Consultant A: Did you enjoy your stay at Kids’ Kingdom? Yes/No.
If yes, please specify your sex: M/F. Consultant B: Did you enjoy your stay at Kids’ Kingdom? Yes/No.
If no, please specify your sex: M /F. Each consultant received 10 completed questionnaires after the survey. The
questionnaires are classiﬁed by the respondents’ answers as follows: Yes No Yes No M F M F
Consultant A Consultant B E] Assume that each respondent of either sex has a probability p of enjoying
his/her stay at Kids’ Kingdom, and that the respondent is a male with prob
ability 1/2. All 20 questionnaires were independently completed. (a) Show that the likelihood function €A(p) based on Consultant A’s data
(5,1,4) is proportional to
116(1 — 104
Write down the corresponding expression for the likelihood function (33 based on Consultant B’s data (6,2,2). [8 marks] (b) Kids’ Kingdom is willing to pay only one consultant for his data. Whose
data would you recommend to Kids’ Kingdom? Explain. [4 marks] (c) If Kids’ Kingdom is willing to pay both consultants for their data, write
down the likelihood function based on the combined data (5, 1, 4, 6, 2, 2). [8 marks] (d) Calculate the Fisher information about p contained in the combined data
in (c). Show that it doubles the Fisher information contained in the data
obtained by either consultant. [10 marks] S&AS: STAT0604 STAT. MODELLING / STAT0605 Math. Stat. 4 3. Let X, Y be independent random variables such that X has an exponential
(A) distribution with density function f(a: I A) = Ae'AI, :1: > 0,
and Y has a Poisson (A) distribution with mass function e‘AAy
9(yl/\)= y, , y=0,1,2,.., where A > 0 is an unknown parameter. The joint probability function of
(X, Y) may be taken to be 1063,31] A)= f($ I AM?! I A)
(a) Write down a likelihood function €0(A) based on (X, Y). [3 marks] (b) Find a sufﬁcient statistic for A.
[5 marks] (0) Show that the Fisher information about A contained in (X, Y) is A+1 = A2 [8 marks] (d) In an experiment, n independent realisations of (X, Y) are obtained: (x11y1)1($21y2)1   v ($11: (i) Find the Fisher information about A contained in the above realisa
tions, and hence determine the CramérRao Lower Bound for A. [6 marks] (ii) Find the maximum likelihood estimate of A based on the n realisa
tions, and describe its large—sample behaviour. [8 marks] S&AS: STAT0604 STAT. MODELLING / STAT0605 Math. Stat. 5 4. Let X1, X2, . . . , Xn be independent random variables with the Pareto density function
f(a:0) = 9(1+ x)“o'1, w > 0. Deﬁne T = — ELI ln(l + Xi).
Suppose we want to test 110:651 against H126>1. (a) Show that the Pareto model has monotone likelihood ratio in T with
respect to H0, H1. [8 marks]
(b) It is known that —T follows the Gamma (n, 0) distribution with density
function 6 1 9
nzn— e I
g($]9)—W, 1‘,0>0. Suppose T = t is observed. Using (a), or otherwise, derive an expression,
in terms of t, for the pvalue of the most powerful test of H0 against H1. [ You may leave your answer in the form of an integral. ]
[8 marks] (c) Suppose n = 3 and it is observed that X1 = 1, X2 = 2 and X3 = 3.
(1) Calculate the observed value of T. [3 marks]
(ii) Calculate the pvalue given in (b). [7 marks]
(iii) Would you reject H0 at the 3.5% signiﬁcance level? [4 marks] ********** ********** S&AS: STAT0604 STAT. MODELLING / STAT0605 Math. Stat. 6 ANNEX
Some useful facts:
1. Distributions
Distribution Probability function, p(y) Mean Variance
. m _
Bin0m1a1(m.p) (y)py(1 p)'" y; mp mp0 —p)
y = 0, 1,. . . , m
e‘AAy
Poisson (A) I ; y = 0,1,2,... A /\
________L________.______
Exponential (A) Ae'Ay; y > 0 1/). 1//\2
onyn—le—Gy Gamma (n,6’) —n1—'—; y > 0 n/O 71/92
_____.____(;L_____—_ 2. Upper quantiles :1: of X2 distributions 0.01 6.6349 9.2104 11.3449 13.2767
0.025 5.0239 7.3778 9.3484 11.1433
0.05 3.8415 5.9915 7.8147 9.4877
0.1 2.7055 4.6052 6.2514 7.7794 3. Generalized likelihood ratio test statistic for testing Ho vs H1: sup H1 Likelihood(6) } A = 21 —————
n {supHo Likelihood(0) 4. Pearson chisquared test statistic: T _ — . . ‘
_ Z T, 0. observed count, E. estimated count. all categories ...
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 Spring '11

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