Nonparametric Estimation from Incomplete Observations

作者：Edward L. Kaplan, Paul Meier · 发表于：Journal of the American Statistical Association · 年份：1958 · DOI：10.1080/01621459.1958.10501452 · 被引用次数：39440 · 研究领域：Statistical Distribution Estimation and Applications、Statistical Methods and Bayesian Inference、Statistical Methods and Inference

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≤t 1ǐ≤t 2ǐ≤ … ≤t N ǐ. Then P(t)= II. [(N – r)/(N – r + 1)], where r assumes those values for which tr ≤t and for which tr ǐ 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...