Abstract: We derive the nonparametric maximum likelihood estimator (NPMLE) of the distribution of the test items using a random, right-censored sample combined with an additional right-censored, residual-lifetime sample in which only lifetimes past a known, fixed time are collected. This framework is suited for samples for which individual test data are combined with left-truncated and randomly censored data from an operating environment. The NPMLE of the survival function using the combined sample is identical to the Kaplan-Meier product-limit estimator only up to the time at which the test items corresponding to the residual sample were known to survive. The limiting distribution for the NPMLE, discussed in detail, leads to confidence bounds for the survival function. For the uncensored case, we study the relative efficiency for the estimator based on the combined sample with respect to the analogous estimator based only on the simple random sample.
Key words and phrases: Counting processes, Nelson-Aalen estimator, relative efficiency, reliability, truncation.