Empirical cumulative lifetime distribution function is often required for selecting lifetime distribution. When some test items are censored from testing before failure, this function needs to be estimated, often via the approach of discrete nonparametric maximum likelihood estimation (DN-MLE). In this approach, this empirical function is expressed as a discrete set of failure-probability estimates. Kaplan and Meier used this approach and obtained a product-limit estimate for the survivor function, in terms exclusively of the hazard probabilities, and the equivalent failure-probability estimates. They cleverly expressed the likelihood function as the product of terms each of which involves only one hazard probability ease of derivation, but the estimates for failure probabilities are complex functions of hazard probabilities. Because there are no closed-form expressions for the failure probabilities, the estimates have been calculated numerically. More importantly, it has been difficult to study the behavior of the failure probability estimates, e.g., the standard errors, particularly when the sample size is not very large. This paper first derives closed-form expressions for the failure probabilities. For the special case of no censoring, the DN-MLE estimates for the failure probabilities are in closed forms and have an obvious, intuitive interpretation. However, the Kaplan–Meier failure-probability estimates for cases involving censored data defy interpretation and intuition. This paper then develops a simple algorithm that not only produces these estimates but also provides a clear, intuitive justification for the estimates. We prove that the algorithm indeed produces the DN-MLE estimates and demonstrate numerically their equivalence to the Kaplan–Meier-based estimates. We also provide an alternative algorithm.
Vulnerability of bridges to scour: insights from an international expert elicitation workshop
