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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 Probability and Statistics II May 5, 2009 Time: 2:30 p.m.  4:30 p.m. Candidates taking examinations that permit the use of calculators may use any
calculator which fulﬁls the following criteria: {a} it should be self—contained, silent, batteryoperated and pocketsized and (b) it should have numeraldisplay facilities
only and should be used only for the purposes of calculation. It is the candidate’s 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 ALL FOUR questions. Marks are shown in square brackets. 1. Let X1, . . . ,Xn be n independent random variables drawn from the density
function (0 + U550: 55 E [021], 0, otherwise, f (3319) = { for some unknown parameter (9 2 0. (a) Show that the moment generating function of the random variable Y = 1n X1 is
M (t) = E [ety] = ii for [t[ suﬂiciently small.
t + 0 + 1’
[4 marks]
(b) Deduce from (a) that
1 1E[lnX1] — — m and Var(lnX1) — (6+ Dz.
[4 marks] (c) Quoting the Central Limit Theorem, show that if 0 = 0, then
ﬁ{ #1711an + 1} converges in distribution to the standard normal distribution N(0, 1) as
n tends to inﬁnity. [4 marks] S&AS: STAT1302 Probability and Statistics II 2 (d) The following questions concern the likelihood ratio test of H0:6=0 against H126>0, based on the observations X1, . . . ,Xn.
(i) Show that the likelihood ratio test statistic is an increasing function
of H]; X1. [4 marks]
(ii) Suppose that n = 1 and X1 is observed to be 0.8. Calculate the
p—value given by the likelihood ratio test. [4 marks] (iii) Suppose that n = 100 and ln Xi is observed to be —80.4. Using (c), calculate an approximate pvalue given by the likelihood
ratio test. [5 marks] ﬂ—upper quantiles of N (0, 1)
0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 ,6 upper
quantile 2.326 2.170 2.054 1.960 1.881 1.812 1.751 1.695 1.645 [Total: 25 marks] 2. The density function of the exponential distribution with mean 0 > 0 is given by
0'1e‘z/9 a: > 0
x 6 = 1 i
f( I > { 0, x S 0_
Let Y1, . . . ,Yn, Z1, . . . ,Zn be independent random variables such that for i =
1, 2, . . . ,n, Y, has the emponential distribution with mean a
and
Z1 has the exponential distribution with mean ab,
for unknown parameters a, b > 0.
Deﬁne )7 = 22:1 YQ/n and Z = ELI Zi/n.
(a) Show that the loglikelihood function of (a, b) is
S'(a, b) = ——n{2lna+lnb+a_1 (17+b’12)}.
[4 marks]
(b) Show that the maximum likelihood estimator of (a, b) is given by
(a,13)=(i7, Z/Y).
[4 marks] S&AS: STAT1302 Probability and Statistics II 3 (c) Show that the Fisher information matrix of (a, b) is “a,” = ] 271(1‘2 'n.(ab)‘1 ] . 'n(ab)"1 nb‘2
[6 marks] ((1) According to large—sample theory, the maximum likelihood estimator
(E1, has an approximate bivariate normal distribution. Specify the
mean vector and the variancecovariance matrix of this bivariate normal
distribution. [6 marks] (e) Suppose that n = 18 and we observe that l7 = 2 and Z = 6. Calculate a twosided Wald conﬁdence interval for b at the 95% conﬁdence
level. [5 marks] ﬁupper quantiles of N(0, 1)
0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 ﬁupper
quantile 2.326 2.170 2.054 1.960 1.881 1.812 1.751 1.695 1.645 [Total: 25 marks] 3. The following data record the monthly numbers of burglaries reported in towns
A and B over a sixmonth period. Town \ Month 123456Tota1
279 10 4 8
18 14 16 12 14 16 28 18 18 19 23 24 The following tabulates some quantiles of the chisquared distribution for your
reference. 5% upper quantiles of X 2
f 1 2 3 4 5 6
upper quantile 3.8415 5.9915 7.8147 9.4877 11.0705 12.5916 (a) Let X. and Y,’ be the numbers of burglaries reported in th_e 2th month in
towns A and B, respectively. Deﬁne X 2 23:1 X1/6 and Y = Z6 Y; / 6. i=1
Suppose that X1, . . . ,X5,Y1,...,Y6 are independent observations, and
that for each 2', X. N Poisson()\A) and K ~ Poisson()\B). S&AS: STAT1302 Probability and Statistics II 4 Note: The Poisson distribution has mean A. Its mass function is —A a:
f(a:)\)=ew')\ forx=0,1,2,....
We wish to test
H0 : AB = 2AA against H1 : AA, AB unrestricted.
(i) Show that the maximum likelihood estimators of AA and AB are
respectively A A _
AA=X and /\B=Y.
[4 marks]
(ii) Show that under H0, the likelihood function is maximized at
1 — _ 2 _ _
[4 marks] (iii) Show that the generalized likelihood ratio test statistic is given by T1: 12 {XinX+?1n(i7/2) — (X+i7)1n [6 marks] (iv) Based on the observed data, would you reject H0 at the 5%
signiﬁcance level using the generalized likelihood ratio test?
[4 marks] (b) Suppose that the 130 burglaries reported in the two towns were committed by 130 independent burglars. Assume that all the burglars
made their Choices of where (i.e. town A or B) and when (i.e. month 1,
2, 3, 4, 5 or 6) to commit burglaries according to the same distribution. Conduct a Pearson chi—squared test at the 5% signiﬁcance level to test
whether a burglar’s choice of where to commit burglary is independent of
his choice of when to commit it. [12 marks] [Total: 30 marks] 4. The table below contains information about the annual inﬂation rates
(averaged over the period 19801989) of 22 countries, where x, = ln(inﬂation rate) for the ith developing country,
y, = ln(inﬂation rate) for the ith developed country, S&AS: STAT1302 Probability and Statistics II 5 for z' = 1,2,. . . , 11. Values of (231 — i)2 and (y1 — g)2 are also provided for
reference, where .7: = 2:1 32/11 and 37 = yi/ll. Developing countries Developed countries Nation 1;. (a2, — 5:)2 Nation M (91' — ylz
COSTA RICA —1.4697 GERMANY 3.5066
BAHAMAS 2.8134 2.2288 FINLAND —2.6593 ZAIRE —0.7985 0.2725 AUSTRALIA 2.5257 0.0176
BARBADOS 2.6593 1.7923 ITALY —2.2073 0.2035
UGANDA —0.3285 0.9840 DENMARK 2.6593 0.0000
URUGUAY 0.7985 0.2725 LUXEMBERG 2.9957 0.1138
TURKEY 0.8916 0.1840 FRANCE 2.6593 0.0000
TANZANIA —1.3093 0.0001 U. KINGDOM 2.6593 0.0000
PERU 0.0770 1.9529 S. AFRICA —1.9661 0.4792
YUGOSLAVIA 0.3147 1.0116 IRELAND 2.4079 0.0627 ETHIOPIA 3.2189 3.6039 BELGIUM —2.9957 0.1138
Total 14.5254 12.3247 Total 29.2421 1.7101
Assume that ($1, . . . ,ccn) and (y1, . . . ,yu) are realisations of two independent
normal random samples, each of size 11, drawn from N (am, 0:) and N011“ of) respectively. (a) Calculate an unbiased estimate of a: and an unbiased estimate of as.
[6 marks] (b) Calculate a 95% equal—tailed conﬁdence interval for 03/03. [10 marks] ﬁupper quantzles of Fag,
_ 0.025 0.050 0.100 0.900 0.950 0.975 a = 10, b = 10 3.717 2.978 2.323 0.431 0.336 0.269
a = 10, b = 11 3.526 2.854 2.248 0.434 0.340 0.273
a = 11, b = 10 3.665 2.943 2.302 0.445 0.350 0.284
a = 11, b = 11 3.474 2.818 2.227 0.449 0.355 0.288 (c) Describe how you would make use of the conﬁdence interval found in (b)
to test
H0 : 03/03 = c against H1 : 03/0: aé c, for some speciﬁed constant c at the 5% level.
What is your conclusion for the test if c = 30? [4 marks] [Total: 20 marks] ...
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 Spring '10
 SMSLee
 Statistics

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