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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT0100 STATISTICAL METHODS
STATO604 STATISTICAL MODELLING (I)
STAT0605 MATHEMATICAL STATISTICS December 30, 2002 Time: 6:30 p.m.  8:30 p.m. Candidates taking examinations that permit the use of calculators may use any cal
culator which fulﬁls the following criteria: (a) it should be selfcontained, silent,
batteryoperated and pocketsized and (b) it should have numeral—display 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 questions. Marks are shown in square brackets. 1. (a) At a shooting competition, a shooter is allowed to ﬁre n rounds from a
naval gun at a ﬁxed target. The ﬁrst round must be ﬁred from a distance
of 100 metres. If the jth round misses the target by 3: metres, then
the (j + 1)th round must be ﬁred from a distance of 1003: metres, for
j: l,2,...,n~—l. Assume that if the shooter ﬁres from a distance of d metres, he will
miss the target by a distance exponentially distributed with the density
function (bl/c0591”, a: > 0, 0, 0:30, f($l9:d) ={ for some unknown parameter 9 > 0. Suppose that the shooter’s jth shot misses the target by X, metres, for
j=1,2,...,n. (i) Show that the likelihood function of 6 based on X1, . . . ,Xn has the
expression 60(6) : 9"exp{—10019(X1 + Xg/X1++ Xn/X 0}. [4 marks] S&AS: STATOlOO Stat. Methods / 0604 Stat. Modelling (I) / 0605 Math. Stat. 2 (ii) Identify, by applying the factorization criterion to 66(6) in (a)(i) or
otherwise, a scalar sufﬁcient statistic for 0.
[2 marks] (b) Suppose that the rule of the competition is modiﬁed such that after ﬁr
ing each round, the shooter has to toss a coin, the outcome of which will
determine the distance from which he will ﬁre the next round. If a head
comes up, the shooter will ﬁre the next round from a distance of 100 me—
tres; otherwise he will ﬁre from a distance of 1003: metres if the previous
round misses the target by 3: metres. The coin is specially designed such
that its head comes up with a probability of 1 — 6—9. The ﬁrst round is
still ﬁred from a distance of 100 metres. The same shooter as in (a) enters this new competition and misses the
target by Yj metres at the jth shot, forj = 1,2, . . . ,n. (i) Show that the likelihood function of 6 based on Y1, . . . , Yn and the
outcomes of the tosses of the coin has the expression n_1 6—9 23:31}
me) = 910.84) < > x 1—69 exp {—100‘16 (Y1 + 3/2/Y111++ Yrs/31311)} , Where I] : 0 if the jth toss of the coin returns a head and = 1 otherwise.
[7 marks]
(ii) Deduce a bivariate sufﬁcient statistic for 9 based on the results in
(NO)
[4 marks] (c) Let n > 1 be ﬁxed. Suppose that at the competition described in (a),
the statistic X1 + X2/X1 +   ' + Xn/Xn1 is observed to be :13; whereas
at the competition described in (b), tails come up at all n — 1 tosses of
the coin and the statistic Y1 + Y2/Y1 +   ‘ + Yn/Yn1 is observed to be
y = x. According to the Likelihood Principle, would you expect to make ex
actly the same inference about 0 based on the data observed from either
competition? Explain. [5 marks]
2. Let X = (X 1, . . . , Xn) constitute a random sample drawn from the density
function
2 a/7r exp {405 — a(a: + ﬂ/$)2}, as > 0,
f (933ml?) =
0, 3? S 0, where a, ﬁ are positive parameters. S&AS: STATOlOO Stat. Methods/0604 Stat. Modelling (I) / 0605 lVIath. Stat. 3 (a) Write down an expression for the loglikelihood function of (0,13) based
on X. [2 marks] (b) If both a and 5 are unknown, ﬁnd the maximum likelihood estimators of
a and ﬂ based on X. [6 marks]
(0) Suppose that ﬂ = 1 and a is unknown. (i) Find the Fisher information about a contained in X. [4 marks]
(ii) Find the maximum likelihood estimator, all say, of (1 based on X. [3 marks]
(iii) Describe the largesample behaviour of 6:1. [4 marks]
(iv) Describe how you would calculate the standard error of (341. [3 marks] (d) Suppose that a = 1 and B is unknown.
(i) Find the maximum likelihood estimator, 31 say, of 5’ based on X. [3 marks]
(ii) Describe how you would calculate the standard error of Bl.
[4 marks]
3. Let X1, . . . , Xn be independent random variables distributed with the common density function
(0—1):c’9, w> 1,
0, cc 3 1, f($19) = { for some unknown parameter 6 > 1. (a) It is known that the moment generating function of the gamma (5, 1)
distribution, for any ,8 > 0, is A/Ig(t) = (1 ~ U”, for t <1. Show that (9 — 1) In H; Xi has the gamma (n, 1) distribution.
[5 marks] (b) Suppose that we wish to test the hypotheses
1110:6360 vs H1:6>60, for a given constant 60 > 1.
(i) Show that the given parametric model has monotone likelihood ratio in some statistic T = T(X1, . . . ,Xn). Specify T.
[5 marks] S&AS: STATOlOO Stat. Methods / 0604 Stat. Modelling (I) / 0605 lVIath. Stat. 4 (ii) Using (a) and (b)(i), suggest a test of size 5%, in terms of the 5%
quantile of the gamma (n, 1) distribution, for testing H0 against H1. [5 marks]
(c) Suppose that we wish to test the hypotheses
H010=60 vs H1:9>1,
for a given constant (90 > 1.
(i) Show that the maximum likelihood estimator of (9 is
ﬁn = 1+n/lnHXi.
i=1
[3 marks] (ii) Describe the generalized likelihood ratio test of size 5% for testing
H0 against H1.
[4 marks]
(d) Using (b)(ii), construct a 95% lower conﬁdence bound for 6.
[4 marks] (e) Using (c) (ii), construct an approximate 95% conﬁdence set for (9. Describe
the shape of this conﬁdence set. [6 marks] 4. In a clinical trial, a group of 8 patients were treated with a new medicine and
their cholesterol levels (in mg/ ml) were recorded one week before and after the
treatment as follows: Patient 1 2 3 4 5 6 7 8
Before treatment 2.8 3.4 2.2 3.7 3.5 3.8 3.0 2.3
2.6 3.3 2.3 3.8 3.0 3.5 2.7 2.2 After treatment Denote by X j and Y} the cholesterol levels of patient j taken one week before
and after the treatment respectively, for j = 1, 2, . . . ,8. (a) Is it appropriate to assume that (X1, , . . ,X8) and (Y1, . . . ,Yg) constitute
two independent random samples? Explain. [3 marks] (b) Assume that
Y] = Xjezv', j=1,2,...,8,
where Z1, . . . , Zg denote 8 independent normal random variables with
common mean ,u and common variance 02, for some unknown parameters
,u E (~oo,oo) and a > 0.
We wish to investigate the effectiveness of the new medicine in lowering
cholesterol level. S&AS: STAT0100 Stat. Methods / 0604 Stat. Modelling (I) / 0605 Math. Stat. 5 (i) Set up an appropriate pair of null and alternative hypotheses, in terms of p, to address the issue. Explain your Choice of hypotheses.
[4 marks] (ii) Conduct a ttest of size 5% for testing the hypotheses set up in (b)(i)
Do you think that the new medicine succeeds in lowering Cholesterol level?
[10 marks] Hint: Upper quantiles a: of Student’s t—distributions
degrees of freedom, f
1P(tf > m) 6 7 8 14 15 16 0.025 2.4469 2.3646 2.3060 2.1448 2.1315 2.1199
0.05 1.9432 1.8946 1.8595 1.7613 1.7531 1.7459
0.1 1.4398 1.4149 1.3968 1.3450 1.3406 1.3368 ...
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 Spring '11

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