Theory III Ph.D Prelim Exam - Summer 2006 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
Page 1 of 2
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
, ,
W n
are iid
N
 
. Let
W n

1 n 
W i
. What is the conditional distribution of  given that n i
 1
W n
 w n what is the conditional distribution of
W n
given  ?) Evaluate the function of w
, m n
   E  |
W n
 w
? (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 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
1 x
2
   x n
 1 are known, and that for two real numbers

1
and 
2
and a c
 we define the function
  
1
 c
 
2
  c

(
Suppose further that (given the parameters
Normal random variables with variance 1, and means
E
Y i
  i
   
Y i
has mean 
1
if parameter vector

,
2
, and c x i
 c , and otherwise has mean
 
2
, c

.
) variables
Y Y Y n
are independent

2
). Consider the statistical problem with

Theory III Ph.D Prelim Exam - Summer 2006 Page 2 of 2 e) Write out a likelihood for this problem, maximize
L n
 c

? Call these 
1
L n
  
2
, c

. For fixed c
, what values of
and  ˆ
2
 
and use the notations

1 and 
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

. In particular, suppose that we
 
2
, c

a (prior) distribution
G
under which the parameters are independent with

1

2


N 0,



2
2


Let g

 | 0,  2

stand for the c

U

 2

probability density and use the notation


Y n
   


 i
 

 1
 i
|  

 g

 | 0,  2
 d
 and


Y n
   



 n 
   1
 i
|  

 g

 | 0,  2
 d
 h) Evaluate E
 
1

Y n
 i) Write (in terms of the functions h
1
and h
2
) a conditional pdf for that this pdf is constant on each interval
 x i
 1
, x i

.) c j) Use your answers to h) and i) to evaluate E
 
1

Y n
 k) For an integer 1 i n evaluate E   
Y Y
2
 Y n
Y Y
2

Y n
. (Notice