ABSTRACT
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 Yn is the process of those “at risk”, then the process Mn defined as
Mn(t) = Nn(t)− ∫ t
0 Yn(s)µ(s)ds, 0≤ t < τ,
is a martingale with respect to the filtration {F nt ,0 ≤ t < τ}, where each σ -algebra is generated by the past of Yn and Nn up to the moment t:
F nt = σ{Yn(s),Nn(s),s≤ t}. (11.1) This was done in the context where Nn(t) counted the number of lifetimes, or failure times, up to the moment t, which happened to be uncensored, while Yn(t) counted the number of those in the initial population of n, who are still under observation and “at risk” of failing.
