=================================================
Assignment 6 Due March 24

1. Given the availability of basic random variables from R
   how can you generate a piecewise exponential random variable?
   
  (To be specific, let us suppose there are 3 pieces, with
    cut points 0 < T_1 < T_2 < T_3= \infty and parameters
                \lambda_1, \lambda_2,  \lambda_3 )


2. Specialize/simplify the Kaplan-Meier estimator and the
   Greenwood formula when all observations are uncensored. Identify
   them with "Sta 291" formula for binomial proportion estimator.

3. (Only for students that have sta635 before:) 
   An estimate of mean survival time, based on the Kaplan-Meier
   estimator, is the following

    \int_0^\infty [ 1- \hat F (t) ] dt  or   \int_0^\infty   t d \hat F(t)

    where \hat F(t) is the Kaplan-Meier estimator.

   Try to derive a variance estimator for the above estimator.
   (it still may be very difficult, but give it a try, and identify
    where is the difficult. 


===========================================================
Assignment 7, Due April 9.

1.  For a two sample testing case, assume there is no censoring
at either samples. 

Simplify the Gehan-Wilcoxon test ( i.e.  sum R1 R2 ( d \hat H1 - d \hat H2 )  )
to identify with the Mann-Whitney test from Sta 621.

Please note, we say two tests are equivalent if the test statistic
they are based upon are a linear function of the other. Since, after
standardization, they will be the same.  For example: tests based
on X and aX + b will be equivalent ( a and b are constants). 

Compare the estimates of variance.

===============================================================
A good review of likelihood based statistical Inference Methods
can be found in Chapter 10 of the following notes. You may want
to read in the next few weeks.

http://www.stat.umn.edu/geyer/old/5102/n2.pdf
===================================================================
A possible project: adopt the R package km.ci and make it up-to-date
and available to the current version R.
=======================================================