THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS
FINAL EXAMINATION
JUNE 2007
MATH3911
HIGHER STATISTICAL INFERENCE
(1) TIME ALLOWED - 2 1/2 Hours
(2) TOTAL NUMBER OF QUESTIONS - 5
(4) THE QUESTIONS ARE NOT OF EQUAL VALUE
(5) TOTAL NUMBER OF MARKS - 125
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
All answers must be written in ink. Except where they are expressly required pencils
may only be used for drawing, sketching or graphical work.
JUNE 2007 MATH3911 Page 2
1. [28 marks] Let X = (Xl, X 2, .. . ,Xn) (n 2: 2) be LLd. random variables
having Poisson distribution with density f(x, A) = e-:/x, x = 0,1,2, ...
a) Show that the product of indicators I{xl=o}(X)·I{xFo}(X) is an unbiased
estimator of the parameter T(A) = e-2A .
n
b) Given that T = L Xi is complete and minimal sufficient for A, derive
i=l
the UMVUE of T(A) = e-2A . (Hint: you may use part (a)).
c) Does the variance of the UMVUE in b) attain the Cramer-Rao bound
for the minimal variance of an unbiased estimator of T(A) = e-2A? Give
d) Suppose now that for the same sample, the parameter of interest is h(A) =
~. Find the MLE of ~, state its asymptotic distribution and, using
this result, suggest a confidence interval for A that asymptotically has
a level 1 - Cl:. Explain why h(A) = ~ is called "variance stabilising
transformation" and why such a transformation it useful.
Hint: For any smooth function h(A) :
/Ti(h().,mle) - h(AO)) ~ N(O, (~~ (Ao))2 I-I (AO)),
8 82
I(A) = E{ 8A [lnf(x, A)]}2 = E{ - 8A2 [lnf(x, A)]}.
e) Let n = 6. Find the uniformly most powerful test of size Cl: = 0.1 for
testing Ho : A ::::: 0.25 against HI : A > 0.25. Justify your answer.
2. [24 marks]
a) A photocopier company kept records of the number of breakdowns per
year for 100 photocopiers, giving the results
Number of breakdowns 0 1 2 3 > 4
Frequency 8 20 37 24 11
Test at the 5% level that the above data comes from a Poisson (A = 2)
distribution.
b) Two companies manufacture white board pens. A sample of 10 pens from
one company and 8 pens from the other were tested until they were no
longer readable. The results (in hours) were:
Company 1 8.9 9.9 12 7.4 6.6 8.4 7.6 10.9 11.7 10.2
Company 2 9.1 13.3 8.2 11.1 10 11.6 10.5 8.7
Using the Wilcoxon Test for large samples, test at the 5% level the hy-
pothesis that there is no difference between the two company's pens.
c) The following table represents data on cortisol level of women in three
different groups: Group 1 delivered their babies by Caesarean section
JUNE 2007 MATH3911 Page 3
,
Group 1 262 307 211 323 454 339 304 154 287 356
Group 2 465 501 455 355 468 362
Group 3 343 772 207 1048 838 687
before the onset of labor, Group 2 delivered by emergency Caesarean
during induced labor and group 3 experienced spontaneous labor.
Use the Kruskal-Wallis test at level 0.05 to test for equality of the average
cortisol level in the three groups.
3. [23 marks] Let X = (Xl, X 2 , .. . ,Xn ) be a sample of size n from a distribution
with density
f(x;O) = [~03/X4, x;::: 0> 0
.. . ., l u elsewhere
a) Prove that the density of X has a monotone likelihood ratio in X(1).
b) Find the cumulative distribution function of Z = X(1). Show that the
density is .
Jz(z;O) = {;::!:~ ,Z;::: 0> 0
o elsewhere
c) Find the uniformly most powerful a-size test rp* of Ho : 0 ::; 2 versus
H l : e> 2 and sketch a graph of Erp* as precisely as possible.
4. [28 marks] In a Bayesian framework, the probability of a defect eE (0,1) of
a certain piece of a machine's output appears to be varying on various days
and can be represented as a random variable with prior density
T(O) = 12e2 (1 - e), e E (0,1).
X. = { 1 if the piece is defective
t 0 if the piece is nondefective
Hence !(Xle) = e'£~=l Xi (1 _ e) n-'£~1 Xi.
On a certain day, n = 16 observations denoted X = (Xl, X 2 ,· .• ,X16 ) are
a) Find the formula for the Bayesian estimator (with respect to quadratic
loss) of the probability e and calculate its estimated value if from the 16
pieces examined on this day, exactly three were defective.
Hint: You may use: B(a, {3) = J~ xa- l (1 - x)f3- l dx, B(a, {3) = ri~~~),
[(a) = Jooo exp( -x)xa-ldx, f(a + 1) = af(a).
b) Find the Bayesian estimator (with respect to quadratic loss ) of the pa-
rameter T/(e) = e(1- e).
c) Using the data of part (a) (3 defective out of 16 tested), test the hypoth-
esis Ho : e ::; 0.25 against H l : e> 0.25 using the Bayesian approach and
a 0-1 loss. Do you accept Ho?
(You may use: Jg.25 e5(1 - e)14de = 0.00000164).
JUNE 2007 MATH3911 Page 4
5. [22 marks] Let X = (Xl, X 2 , . .. ,Xn ) be i.i.d., each with uniform [0,8) density
(8) 0).
a) Find COV(X(n_l), X(n)).
b) Find the density of the range R = X(n) - X(1). Using it, or otherwise,
show that ER = ~~~ 8 holds.
Some useful formulae
1) Wilcoxon: Two independent samples Xl, X 2,· . . , X m and Y1 , Y2, ... ,Yn ,
W = ,,~ R(X). Then W m +n -m(m+n+l)/2 has approximately a stan-
m+n L..~_l ~ y'mn(m+n+l)/12
dard normal distrihution.
2) The Kruskal-Wallis statistic for a K sample problem:
12 K _ n + 1 2 12 K RT.
H = L ni(Ri. - -) = L - - 3(n + 1)
n(n + 1) i=l 2 n(n + 1) i=l ni
has Xk-l as a limiting distribution under the null hypothesis.
3) The rth order statistic (r = 1,2,. " ,n) of the sample of size n from a
distribution with a density !X(.) and a cdf Fx (.) :
The joint density of the pair (X(i)' X(j)), 1 :S i < j :S n is:
!x(i),x(j) (x, y) =
(i - l)!(j - ~!_ i)!(n _ j)! !X (x)!x (y)[Fx (xW-1 [Fx (y)-Fx (x)jJ-I-i[l-Fx (y)]n- j
for x < y.
4) Bayesian inference:
h(8IX) = !(~~i~(8), g(X) = le !(XI8)T(8)d8. 