Theory III Ph.D Prelim Exam  Summer 2006 Page 1 of 2 a)
Prove the following simple lemma. (You may use the lemma in what follows even if you can not prove it.).
Lemma
Suppose that
F
is a continuous distribution with probability density function on
C
exp
ax
2
bx
(for real numbers
a
0, , and
b
). Then
F
is Normal with mean
b
/ 2
a
and variance 2 1/ 2
a
.
b)
Suppose that is a random vector such that
T
2 and that conditioned on
T
t
,
U
. Find the conditional distribution of
T
given that
U
u
.
c)
Now suppose that 2 and that conditioned on , variables , 2 N . Let
W n
1
n i n
1
W i
. What is the conditional distribution of given that
W n
what is the conditional distribution of
W n
given ?) Evaluate the function of
w
,
m n
E 
W n
w
W n
are iid
w n
? (Hint:
d)
Now suppose that N . With
m n
is a fixed unknown quantity and that the variables as defined in
c)
consider the random quantity
m W n
, 2
W n
are iid . Show that this converges to a constant in probability and identify the limit. Now suppose that
n
values 0
x
2
x n
1 are known, and that for two real numbers 1 and 2 and a
c
we define the function 1 2
c
Suppose further that (given the parameters , 2 , and
c
) variables
Y n
are independent Normal random variables with variance 1, and means E
Y i
i
(
Y i
has mean 1 parameter vector if
x i
, 2
c
, and otherwise has mean ,
c
. 2 ). Consider the statistical problem with
Theory III Ph.D Prelim Exam  Summer 2006 e)
Write out a likelihood for this problem,
L n
, 2 ,
c
. For fixed
c
Page 2 of 2
, what values of 1 and maximize
L n
c
? Call these 1 and ˆ 2 and use the notations 2  for the N
n
i
1
i
c
f)
Is there a unique maximum likelihood estimator for the parameter vector , 2 ,
c
? Explain carefully.
g)
As explicitly as is possible, give a likelihood ratio test statistic for testing the hypothesis H :
c
.5
versus H :
c
.5
. Consider a Bayes version of the inference problem for give , 2 ,
c
1 2 N 0, 2 2 , 2 ,
c
. In particular, suppose that we a (prior) distribution
G
under which the parameters are independent with Let
g
 0, 2 stand for the 2 2
Y n
Y n
2
c
U
probability density and use the notation
i
1
n
1
i

i g

g
 0, 2  0,
d
and 2
d
h)
Evaluate E 1 2
Y n
i)
Write (in terms of the functions
h
1 and
h
2 that this pdf is constant on each interval
x i
1 ) a conditional pdf for ,
x i
.)
c
j)
Use your answers to
h)
and
i)
to evaluate E 1 2
Y n
k)
For an integer 1
n
evaluate E
Y Y
2
Y n
2
Y n
. (Notice