Homework 4 . Due Oct. 7, 2016 


(1). Verify for all t

               [1-\hat F_{KM} (t)]*[1-\hat G (t)] = #(T_i > t)/n

where 
       1-\hat G(t) = \prod_{s <= t} [ 1 - dM(s)/ R(s)]
and 
       M(s) = #(T_i <= s, \delta_i =0).


(2). We have defined the Kaplan-Meier estimator and the Greenwood formula


    1- \hat F_{KM} (t) = \prod_{ s<= t}  [ 1 - dN(s)/ R(s) ] ;

    Greenwood:  [1-\hat F_{KM}(t)]^2 \int_0^t \frac{dN(s)}{R(s)[R(s) - dN(s)]} . 


    Now suppose there is no censoring before time T (i.e. all censoring ocure after T).
    Simplyfy the Greenwood formula [for variance of Kaplan-Meier estimator 1- \hat F_{KM}(T)] to 
    the usual variance estimator of binomial MLE of p.



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