
Lecture 11
Statistical inference about F,
based on the Kaplan–Meier
estimator
Before we consider three specific statistical problems let us quickly
look back on what was done in Lectures 9 and 10. The key model
derived there was that if we have censored observations, and if Y
n
is
the process of those “at risk”, then the process M
n
defined as
M
n
(t) = N
n
(t) −
Z
t
0
Y
n
(s)µ(s)ds, 0 ≤t < τ,
is a martingale with respect to the filtration {F
n
t
,0 ≤ t < τ}, where
each σ-algebra is generated by the past of Y
n
and N
n
up to the moment
t:
F
n
t
= σ {Y
n
(s),N
n
(s),s ≤t}. (11.1)
This was done in the context where N
n
(t) counted the number of life-
times, or failure times, up to the moment ...