unpleasant property of this estimator, that suggests that perhaps it is not the "best" estimator, is that it ignores all the observations whose censoring time precedes . Intuitively, these observations still contain information about : For example, when for many events with , also holds, we can infer that events often happen early, which implies that is large, which, through means that must be small. However, this information is ignored by this naive estimator. The question is then whether there exists an estimator that makes a better use of all the data. This is what the Kaplan–Meier estimator accomplishes. Note that the naive estimator cannot be improved when censoring does not take place; so whether an improvement is possible critically hinges upon whether censoring is in place. The plug-in approach By elementary calculations, where the second to last equality used that is integer valued and for the last line we introduced By a recursive expansion of the equality , we get Note that here . The Kaplan–Meier estimator can be seen as a "plug-in estimator" where each is estimated based on the data and the estimator of is obtained as a product of these estimates. It remains to specify how is to be estimated. By Proposition 1, for any such that , and both hold. Hence, for any such that , By a similar reasoning that lead to the construction of the naive estimator above, we arrive at the estimator (think of estimating the numerator and denominator separately in the definition of the "hazard rate" ). The Kaplan–Meier estimator is then given by The form of the estimator stated at the beginning of the article can be obtained by some further algebra. For this, write where, using the actuarial science terminology, is the number of known deaths at time , while is the number of those persons who are alive (and not being censored) at time . Note that if , . This implies that we can leave out from the product defining all those terms where . Then, letting be the times when , and , we arrive at the form of the Kaplan–Meier estimator given at the beginning of the article: As opposed to the naive estimator, this estimator can be seen to use the available information more effectively: In the special case mentioned beforehand, when there are many early events recorded, the estimator will multiply many terms