Prove the following statements in Problem 1 and 2.

Problem 1 

Assume that UMVUE and MLE exist.  Show that they both depend on
data only through a sufficient statistic.

                                                                                
Problem 2
                                                                                
If a Bayes estimator is unique, i.e., any two estimators
that minimizes the Bayes risk are equal with probability one for
any given parameter, then it is admissible.
                                                                                
Problem 3
                                                                                
a)Assume that Xi are n  i.i.d normally distributed with mean theta and 
standard deviation sigma.  Also assume that theta has a normal prior 
distribution with mean mu and standard deviation tao.  Find the posterior 
distribution of theta given Xi's while assuming that sigma, mu and tao are 
all known.
                                          
For b) and c) below, consider the squared error loss.                                      
b) Consider a linear estimator c*sum(Xi)/n.  For 0<c<1, prove that
the estimator is admissible by using a) and a theorem (called theorem *
in lectures).   The major calculation to be done is the Bayes risk.


c) What if c=0?  Is the liner estimator admissible?  How about c is
less than zero?  Prove your assertions.


Additionally, please do excercises 5,11,12 in Chpapter 4 of Berger
See: 
http://www.math.cornell.edu/~agzhao/math674_08spring/HW4_2.pdf