Nonparametric Estimation from Incomplete Observations
Abstract: In lifetesting, medical follow-up, and other fields the observation of the time of occurrence of the event of interest (called a death) may be prevented for some of the items of the sample by the previous occurrence of some other event (called a loss). Losses may be either accidental or controlled, the latter resulting from a decision to terminate certain observations. In either case it is usually assumed in this paper that the lifetime (age at death) is independent of the potential loss time; in practice this assumption deserves careful scrutiny. Despite the resulting incompleteness of the data, it is desired to estimate the proportion P(t) of items in the population whose lifetimes would exceed t (in the absence of such losses), without making any assumption about the form of the function P(t). The observation for each item of a suitable initial event, marking the beginning of its lifetime, is presupposed. For random samples of size N the product-limit (PL) estimate can be defined as follows: L...
Cites background from "Nonparametric Estimation from Incom..."
...The functions (7) and (8) are maximum-likelihood estimates in the family of all possible distributions (Kaplan and Meier, 1958)....
...…A(u)du}= 1I7I j1-i (8) u aO t(i)K (i For uncensored ata this is the usual sample survivor function; some of the asymptotic properties of (8) are given by Kaplan and Meier (1958) and by Efron (1967) and can be used to adapt to the censored case tests based on sample cumulative distribution function....
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