Template:Kaplan-meier estimator

Kaplan-Meier Estimator
The Kaplan-Meier estimator, also known as the product limit estimator, can be used to calculate values for nonparametric reliability for data sets with multiple failures and suspensions. The equation of the estimator is given by:


 * $$\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\frac{{{n}_{j}}-{{r}_{j}}},\text{ }i=1,...,m$$

where:


 * $$\begin{align}

& m= \text{the total number of data points} \\ & n= \text{the total number of units} \end{align}$$

The variable $${{n}_{i}}$$  is defined by:


 * $${{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m$$

where:


 * $$\begin{align}

& {{r}_{j}}= \text{the number of failures in the }{{j}^{th}}\text{ data group} \\ & {{s}_{j}}= \text{the number of suspensions in the }{{j}^{th}}\text{ data group} \end{align}$$

Note that the reliability estimate is only calculated for times at which one or more failures occurred. For the sake of calculating the value of $${{n}_{j}}$$  at time values that have failures and suspensions, it is assumed that the suspensions occur slightly after the failures, so that the suspended units are considered to be operating and included in the count of  $${{n}_{j}}$$.

Example 1: