Practical alternatives for estimating the failure probabilities of censored life data

For censored life data, Kapur and Lamberson and O'Connor recommend the use of Johnson's formula for non‐parametric estimation of the failure distribution, F ( t ). The formula is used to calculate the adjusted ranks of the recorded failures, which are input into the median rank estimation equation of F ( t ). It is our experience that Johnson's formula is fairly difficult for the reliability practitioner to understand and implement. Fortunately, an alternative formula has been developed which is much easier to use. It is demonstrated that the calculated adjusted ranks may be used in either the mean rank or median rank equations for the estimation of F ( t ). The question which we pose is the following: How does the performance of Johnson's estimator compare with that of the more commonly known and understood Kaplan‐Meier, or product‐limit, estimator?' To answer this question, the Kaplan‐Meier procedure is evaluated with respect to its equivalent adjusted rank of recorded failures. The two procedures are determined to be equivalent with respect to adjusted rank criteria. Therefore, it is proved that with Johnson's estimator adapted for use with the mean rank estimator, the two procedures will yield identical estimates of the failure probabilities. Based upon this finding, it is our recommendation thatthe reliability practitioner use the alternative formula for generation of the adjusted ranks, followed by use of either the mean or median rank formula.