8. Assessing Product Reliability 8.2. Assumptions/Prerequisites 8.2.1. How do you choose an appropriate life distribution model? 8.2.1.5. Empirical model fitting  distribution free (KaplanMeier) approach 

The KaplanMeier procedure gives CDF estimates for complete or censored sample data without assuming a particular distribution model  The KaplanMeier (KM) Product
Limit procedure provides quick, simple estimates of the
Reliability function
or the CDF
based on failure data that may even be
multicensored.
No underlying model (such as Weibull or lognormal) is assumed; KM estimation
is an empirical (nonparametric) procedure. Exact times of failure are
required, however.
Calculating Kaplan  Meier Estimates The steps for calculating KM estimates are the following:

Example of KM estimate calculations 
A simple example will illustrate the KM procedure.
Assume 20 units are on life test and 6 failures occur at the following
times: 10, 32, 56, 98, 122, and 181 hours. There were 4 unfailed units
removed from the test for other experiments at the following times: 50
100 125 and 150 hours. The remaining 10 unfailed units were removed from
the test at 200 hours. The KM estimates for this life test are:
\(R\)(10) = 19/20
A General Expression for KM Estimates A general expression for the KM estimates can be written. Assume we have \(n\) units on test and order the observed times for these \(n\) units from \(t_1\) to \(t_n\). Some of these are actual failure times and some are running times for units taken off test before they fail. Keep track of all the indices corresponding to actual failure times. Then the KM estimates are given by: $$ \hat{R}(t_i) = \prod_{\begin{array}{c} j \in S \\ t_j \le t_i \end{array}} \frac{nj}{nj+1} \, , $$ with the "hat" over \(R\) indicating it is an estimate and \(S\) is the set of all subscripts \(j\) such that \(t_j\) is an actual failure time. The notation \(j \in S\) and \(t_j\) less than or equal to \(t_i\) means we only form products for indices \(j\) that are in \(S\) and also correspond to times of failure less than or equal to \(t_i\). Once values for \(R(t_i)\) are calculated, the CDF estimates are \(F(t_i) = 1  R(t_i)\). 
A small modification of KM estimates produces better results for probability plotting 
Modified KM Estimates
The KM estimate at the time of the last failure is \(R(t_r)\) = 0 and \(F(t_r)\) = 1. This estimate has a pessimistic bias and cannot be plotted (without modification) on a probability plot since the CDF for standard reliability models asymptotically approaches 1 as time approaches infinity. Better estimates for graphical plotting can be obtained by modifying the KM estimates so that they reduce to the median rank estimates for plotting Type I Censored life test data (described in the next section). Modified KM estimates are given by the formula $$ \hat{R}(t_i) = \frac{n + 0.7}{n + 0.4} \prod_{\begin{array}{c} j \in S \\ t_j \le t_i \end{array}} \frac{nj+0.7}{nj+1.7} \, . $$ Once values for \(R(t_i)\) are calculated, the CDF estimates are \(F(t_i) = 1  R(t_i)\). 