#   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)

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