the Kaplan-Meier estimator

Suppose you follow a group of patients (or insured lives, or machines) and want to estimate the survival function directly from the data — without assuming any formula. A real complication is that you rarely observe everyone until they die: some are still alive when the study ends, some drop out early. These incompletely observed lives are called censored. The Kaplan-Meier estimator is the standard, distribution-free way to build a survival curve from exactly this kind of messy, partly-censored data.

It works by stepping through the distinct times at which deaths actually occur and chaining together the survival odds. At each death time you ask: of those still under observation and at risk just before, what fraction survived past this instant? That fraction is one minus deaths-over-at-risk at that moment. The estimated survival curve is the running product of all those fractions up to a given time — which is why it is also called the product-limit estimator. Censored lives count toward the at-risk pool only up to the moment they drop out, so they reduce later denominators without being treated as deaths. The result is a staircase that steps down only at observed death times.

Kaplan-Meier is the workhorse of clinical trials, reliability testing, and any survival study where you cannot wait for everyone to die — it is how 'median survival' and 'five-year survival' get reported. For actuaries it is a non-parametric companion to a graduated life table: useful when you want the data to speak for itself, with the honest caveat that the right-hand tail of the curve, based on very few survivors, is shaky, and that it assumes censoring is unrelated to a person's underlying risk.