Homework three. Due Oct. 2, 2009

1. Suppose we have a random sample of right censored observations:
   (T_i, delta_i) = (times, status) = ( \min( Y_i, C_i) , \delta[Y_i <= C_i] )
   where we assume Y_i is from a piecewise exponential distribution.
 

   Derive the MLE for lambda_j for the k piece piece-wise exponential
   distribution based on n right censored data.
   Assume lambda_j is for the jth piece and it is for the 
   interval [tau_{j-1} , tau_j ). 



   Derive the Fisher information matrix and its inverse matrix for the 
   MLE of lambda_j, j=1,...,k for the k piece 
   piece-wise exponential
   distribution based on n right censored data, i.e. (T_i, \delta_i )
   for i=1,2,...n  [assume n >> k]



2. [due Oct. 5] Specialize/simplify the Kaplan-Meier estimator, when all status
   indicators are identically equal to one, i.e. no censor, to the
   sample proportion.

   Similarly simplify the
   Greenwood formula when all observations are uncensored to the
   "Sta 291" formula for the variance of binomial proportion estimator.

   Someone told me the variance formula can only simplify to "Sta 291"
   formula when there is no tie in the survival times. I honestly do not
   remember if this is the case. So I ask you to investigate if this is true.


Homework four.


Show that if F1(t) and F2(t) satisfy the relation  (for some constant a >0 )

                     1-F1(t) = [ 1-F2(t)]^a   for all  t

then the hazards of the 2 populations satisfy "proportional hazards".
(assume both hazards exist).