NATIONAL UNIVERSITY OF SINGAPORE EXAMINATION ST2132 ...
NATIONAL UNIVERSITY OF SINGAPORE
EXAMINATION
ST2132
MATHEMATICAL STATISTICS
(Semester 1: AY 2011-2012)
November 2011 -
Time Allowed: 2 Hours
INSTRUCTIONS TO CANDIDATES 1. This examination paper contains SEVEN (7) questions and comprises ELEVEN (11)
printed pages. 2. Answer ALL questions. Except for Question 1, show your work, quoting from page 11 wherever applicable. Page 10 is for rough work.
3. This is a CLOSED book examination. Candidates may bring in a programmable calculator. 4. An A4 size help sheet written on both sides is allowed.
Matriculation No: _ _ _ _ _ _ _ _ _ _ _ Seat No: _ _ _ _ __
Question
1
2
3
4
5
6
7
Max. marks
1
4
6
23
10
10
26
Marks scored
1
I
Total
80
ST2132 1. [1 mark] The unconditional probability of event A is 1/3; the unconditional probability of B is 1/2. If A and B are independent, they must also be mutually exclusive. (A) True
(B) False
2. [4 marks] Let x be the realisation of one random draw from a population with mean J-L and SD a, both unknown. Say whether the statement is true or false, and explain briefly. (a) [2 marks] x is an unbiased estimate of J-L.
(b) [2 marks] The exact SE is O.
3. [6 marks] Let XI, ... ,Xn be IID N(J-L,a 2 ) random variables, where J-L and a 2 are unknown
=
constants. Let
n
-1 .z:::r=l Xi.
The maximum likelihood estimator of a 2 ,
fj2,
is exactly
X~-l' Theory says its approximate distribution for large n is N (a 2 , 2~4). Show that the exact and approximate distributions are similar for large n. You may use the
distributed as
n
following: (i) A
X
xi
random variable has expectation 1 and variance 2. (ii) A x~ random
variable has the same distribution as the sum of k lID
2
xi random variables.
ST2132
4. [23 marks] Let Xl,'" ,Xn be IID with density
f(xl>') where>' > O. You are given that E(XI)
>.2 xe -)'x,
= 2/>.
x> 0
and var(XI}
2/>.2.
(a) [2 marks] Find the method of moments estimator of >..
(b) [4 marks] Find the maximum likelihood estimator .\ of >..
(c) [3 marks] Show that the Fisher information in Xl for>' is 2/>.2.
(d) [2 marks] Find an asymptotic distribution for .\.
3
ST2132 (e) [3 marks) 25 realisations of the X's have an average of 1.81. Calculate the ML estimate of A. Use (d) to compute an approximate SE for your estimate.
(f) [6 marks) Someone thinks the sample size is too small for the SE in (e) to be reliable, and is also concerned about bias. How can you investigate these issues numerically? Give details comprehensible to a person who knows how to generate realisations from the X's.
(g) [3 marks) If the instructions in (f) can be used to calculate the efficiency of ,\ approxi mately, outline the steps. If not, explain briefly.
4
ST2132
5. [10 marks] Let Xl, ... , Xn be lID Gamma(\ a) random variables. The density is
f(xla, A) (a) [4 marks] Show that Tl a.
A''' a-I -AX f(a)x e ,
I:f,.-d Xi
and T2
= I1f=1 Xi
(b) [3 marks] Given that E(Xd = III = a/A and var(Xl) method of moments (MOM) estimators are
x>o are sufficient statistics for A and
= 112
Ili
= a/A2 , show that the
&2'
(c) [3 marks] Is j a function of Tl and T2? What can you say about E(~ITt, T 2 ), without calculating it?
5
ST2132
6. [10 marks] Let Xl, ... ,Xn be UD N(/.L, (T2), where and the alternative hypothesis is /.L
(T
is known. Suppose the null hypothesis is /.L
/.Ll, where 111
< 110·
(a) [5 marks] The likelihood ratio is
£(110) = exp {_ 2nx(111 -110) n(115 £(111) 2(T2
-l1r)}
Show that this yields critical regions of the form {x < c} for some constant c.
(b) [3 marks] For a E (0,1), consider a test of size a of the form in (a). Show that
c where P(Z > ze.)
(T
110 - Ze. Vii
a for the standard normal Z.
(c) [2 marks] Show that the power of the test in (b) is
6
= /.Lo
ST2132
7. [26 marks] Let N = (N1 ,N2,N3,N4 )
'"
Multinomial(n,p), i.e.
)pnlpn2 pn3pn4, n f(nlp) ( nl nz n3 n4 1 2 3 4
nES
(a) [2 marks] What is S?
(b) [2 marks] N is the sum of n lID random vectors WI, ... , W n. Write down the complete density function of W 1.
(c) [6 marks] Show that the maximum likelihood estimator of pis Nl,Nz,N3,N4 ) ( n n n n
7
ST2132 (d) [6 marks] In a certain genetic experiment, there is a
P
=
«(J)
P
=
(2 + 1- 1(J
4
(J
'
4
'
Show that if 0 is the maximum likelihood estimator of
(e) [2 marks] Let
(J E (J
4
(J,
(0,1) such that
~)
'4
then
n denote the set of all positive probability vectors of length 4.
the goodness-of-fit of the model in (d), a statistician states Ho: P E wo, What is wo?
8
To check
ST2132
(f)
[4 marks] Let the likelihood function be 4
L(p)
IIpfi i=l
Show that maxL(p) pEwo
i=l
IT (~i)Ni
maxL(p) pEn
t=l
(g) [2 marks] What is the null distribution of the likelihood ratio test statistic, for large n?
(h) [2 marks] Is the word "null" important in (g)?
9
ST2132
[FOR ROUGH WORK]
10
Let X!, ... Xn be IID random variables with expectation 1
and if
2
n
-1
n
(
Li=l Xi
{t
and SD
IJ.
Let
X=
ST2132 Xi
n- 1
- 2
X).
X . . . . {t.
(1) The Law of Large Numbers: as n ........
00,
(2) The Central Limit Theorem: as n ........
00,
Xl
+ ... + Xn
-----;:::---'-
........ N(O , 1)
(3) Let X have density J(xIO), 0 E 8. The Fisher information is
I(O) = -E [::210gJ(XI0)]
(4) Let
0 be an
(5) Let
Bbe a consistent
unbiased estimator of O. Then for every 0 E 8, var(O) ?:: I(O)-ljn. estimator of O. Its efficiency is (I(O)-ljn)jvar(B).
Let Xl>'" ,Xn be IID with density J(xIO), 0 E 6.
(6) Let fJ be the maximum likelihood estimator of O. Under certain conditions, as n ........
yfnI(O)(B - 0)
00,
N(O,l)
(7) T(Xt, ... , Xn) is sufficient for 0 if there are functions get, 0) and hex) such that for every
oE 8 J(xIO) = g(t(x), 0) hex),
all possible x
(8) The Rao-Blackwell theorem: Let fJ be an estimator of 0, and T be sufficient for O. Then the mean square error of E(OIT) is smaller than that of 0, unless 0 is a function of T.
(9) Let a test of Ho : 0 = 00 against HI : 0
01 have critical region {x ERe ~n}. The size is
Po(X E R), the power is Pl(X E R). Let the random likelihood function be L(O; X)
= J(X1 10)··. J(XnIO).
(lO) The Neyman-Pearson Lemma: Among all tests of size:::; critical region of the form L(OO;X) { L(Ol;X) (11) Let wo c n be subsets of 8. The likelihood ratio is
0:,
the most powerful one has
EXAMINATION
ST2132
MATHEMATICAL STATISTICS
(Semester 1: AY 2011-2012)
November 2011 -
Time Allowed: 2 Hours
INSTRUCTIONS TO CANDIDATES 1. This examination paper contains SEVEN (7) questions and comprises ELEVEN (11)
printed pages. 2. Answer ALL questions. Except for Question 1, show your work, quoting from page 11 wherever applicable. Page 10 is for rough work.
3. This is a CLOSED book examination. Candidates may bring in a programmable calculator. 4. An A4 size help sheet written on both sides is allowed.
Matriculation No: _ _ _ _ _ _ _ _ _ _ _ Seat No: _ _ _ _ __
Question
1
2
3
4
5
6
7
Max. marks
1
4
6
23
10
10
26
Marks scored
1
I
Total
80
ST2132 1. [1 mark] The unconditional probability of event A is 1/3; the unconditional probability of B is 1/2. If A and B are independent, they must also be mutually exclusive. (A) True
(B) False
2. [4 marks] Let x be the realisation of one random draw from a population with mean J-L and SD a, both unknown. Say whether the statement is true or false, and explain briefly. (a) [2 marks] x is an unbiased estimate of J-L.
(b) [2 marks] The exact SE is O.
3. [6 marks] Let XI, ... ,Xn be IID N(J-L,a 2 ) random variables, where J-L and a 2 are unknown
=
constants. Let
n
-1 .z:::r=l Xi.
The maximum likelihood estimator of a 2 ,
fj2,
is exactly
X~-l' Theory says its approximate distribution for large n is N (a 2 , 2~4). Show that the exact and approximate distributions are similar for large n. You may use the
distributed as
n
following: (i) A
X
xi
random variable has expectation 1 and variance 2. (ii) A x~ random
variable has the same distribution as the sum of k lID
2
xi random variables.
ST2132
4. [23 marks] Let Xl,'" ,Xn be IID with density
f(xl>') where>' > O. You are given that E(XI)
>.2 xe -)'x,
= 2/>.
x> 0
and var(XI}
2/>.2.
(a) [2 marks] Find the method of moments estimator of >..
(b) [4 marks] Find the maximum likelihood estimator .\ of >..
(c) [3 marks] Show that the Fisher information in Xl for>' is 2/>.2.
(d) [2 marks] Find an asymptotic distribution for .\.
3
ST2132 (e) [3 marks) 25 realisations of the X's have an average of 1.81. Calculate the ML estimate of A. Use (d) to compute an approximate SE for your estimate.
(f) [6 marks) Someone thinks the sample size is too small for the SE in (e) to be reliable, and is also concerned about bias. How can you investigate these issues numerically? Give details comprehensible to a person who knows how to generate realisations from the X's.
(g) [3 marks) If the instructions in (f) can be used to calculate the efficiency of ,\ approxi mately, outline the steps. If not, explain briefly.
4
ST2132
5. [10 marks] Let Xl, ... , Xn be lID Gamma(\ a) random variables. The density is
f(xla, A) (a) [4 marks] Show that Tl a.
A''' a-I -AX f(a)x e ,
I:f,.-d Xi
and T2
= I1f=1 Xi
(b) [3 marks] Given that E(Xd = III = a/A and var(Xl) method of moments (MOM) estimators are
x>o are sufficient statistics for A and
= 112
Ili
= a/A2 , show that the
&2'
(c) [3 marks] Is j a function of Tl and T2? What can you say about E(~ITt, T 2 ), without calculating it?
5
ST2132
6. [10 marks] Let Xl, ... ,Xn be UD N(/.L, (T2), where and the alternative hypothesis is /.L
(T
is known. Suppose the null hypothesis is /.L
/.Ll, where 111
< 110·
(a) [5 marks] The likelihood ratio is
£(110) = exp {_ 2nx(111 -110) n(115 £(111) 2(T2
-l1r)}
Show that this yields critical regions of the form {x < c} for some constant c.
(b) [3 marks] For a E (0,1), consider a test of size a of the form in (a). Show that
c where P(Z > ze.)
(T
110 - Ze. Vii
a for the standard normal Z.
(c) [2 marks] Show that the power of the test in (b) is
6
= /.Lo
ST2132
7. [26 marks] Let N = (N1 ,N2,N3,N4 )
'"
Multinomial(n,p), i.e.
)pnlpn2 pn3pn4, n f(nlp) ( nl nz n3 n4 1 2 3 4
nES
(a) [2 marks] What is S?
(b) [2 marks] N is the sum of n lID random vectors WI, ... , W n. Write down the complete density function of W 1.
(c) [6 marks] Show that the maximum likelihood estimator of pis Nl,Nz,N3,N4 ) ( n n n n
7
ST2132 (d) [6 marks] In a certain genetic experiment, there is a
P
=
«(J)
P
=
(2 + 1- 1(J
4
(J
'
4
'
Show that if 0 is the maximum likelihood estimator of
(e) [2 marks] Let
(J E (J
4
(J,
(0,1) such that
~)
'4
then
n denote the set of all positive probability vectors of length 4.
the goodness-of-fit of the model in (d), a statistician states Ho: P E wo, What is wo?
8
To check
ST2132
(f)
[4 marks] Let the likelihood function be 4
L(p)
IIpfi i=l
Show that maxL(p) pEwo
i=l
IT (~i)Ni
maxL(p) pEn
t=l
(g) [2 marks] What is the null distribution of the likelihood ratio test statistic, for large n?
(h) [2 marks] Is the word "null" important in (g)?
9
ST2132
[FOR ROUGH WORK]
10
Let X!, ... Xn be IID random variables with expectation 1
and if
2
n
-1
n
(
Li=l Xi
{t
and SD
IJ.
Let
X=
ST2132 Xi
n- 1
- 2
X).
X . . . . {t.
(1) The Law of Large Numbers: as n ........
00,
(2) The Central Limit Theorem: as n ........
00,
Xl
+ ... + Xn
-----;:::---'-
........ N(O , 1)
(3) Let X have density J(xIO), 0 E 8. The Fisher information is
I(O) = -E [::210gJ(XI0)]
(4) Let
0 be an
(5) Let
Bbe a consistent
unbiased estimator of O. Then for every 0 E 8, var(O) ?:: I(O)-ljn. estimator of O. Its efficiency is (I(O)-ljn)jvar(B).
Let Xl>'" ,Xn be IID with density J(xIO), 0 E 6.
(6) Let fJ be the maximum likelihood estimator of O. Under certain conditions, as n ........
yfnI(O)(B - 0)
00,
N(O,l)
(7) T(Xt, ... , Xn) is sufficient for 0 if there are functions get, 0) and hex) such that for every
oE 8 J(xIO) = g(t(x), 0) hex),
all possible x
(8) The Rao-Blackwell theorem: Let fJ be an estimator of 0, and T be sufficient for O. Then the mean square error of E(OIT) is smaller than that of 0, unless 0 is a function of T.
(9) Let a test of Ho : 0 = 00 against HI : 0
01 have critical region {x ERe ~n}. The size is
Po(X E R), the power is Pl(X E R). Let the random likelihood function be L(O; X)
= J(X1 10)··. J(XnIO).
(lO) The Neyman-Pearson Lemma: Among all tests of size:::; critical region of the form L(OO;X) { L(Ol;X) (11) Let wo c n be subsets of 8. The likelihood ratio is
0:,
the most powerful one has
