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: List and label the N observed lifetimes (whether to death or loss) in order of increasing magnitude, so that one has $0 \leqslant t_1^\prime \leqslant t_2^\prime \leqslant \cdots \leqslant t_N^\prime .$ Then $\hat P\left( t \right) = \Pi r\left[ {\left( {N - r} \right)/\left( {N - r + 1} \right)} \right]$, where r assumes those values for which $t_r^\prime \leqslant t$ and for which $t_r^\prime$ measures the time to death. This estimate is the distribution, unrestricted as to form, which maximizes the likelihood of the observations.

Other estimates that are discussed are the actuarial estimates (which are also products, but with the number of factors usually reduced by grouping); and reduced-sample (RS) estimates, which require that losses not be accidental, so that the limits of observation (potential loss times) are known even for those items whose deaths are observed. When no losses occur at ages less than t the estimate of P(t) in all cases reduces to the usual binomial estimate, namely, the observed proportion of survivors.

Prepared while the authors were at Bell Telephone Laboratories and Johns Hopkins University respectively. The work was aided by a grant from the Office of Naval Research.