Abstract
This paper presents an aggregate method of selecting a theoretical cumulative distribution function (CDF) for an empirical CDF. The method was intended to identify the time of reliable operation of a renewable technical object by applying three criteria based on the following statistics: the modified Kolmogorov–Smirnov (MK-S) statistic, the mean absolute deviation of the theoretical CDF from the empirical CDF, and a statistic calculated on the basis of a log-likelihood function. The values of these statistics were used to rank eleven probability distributions. The data for which calculations were made concerned failures of the driver’s cab lock recorded during five years of operation of a fleet of 45 trams. Before calculating the statistics, the empirical CDF of the examined component was determined using the Kaplan–Meier estimator, and then, using the method of Maximum Likelihood Estimation, the parameters of the analysed theoretical distributions were estimated. The theoretical distributions were then ranked according to the values obtained for each of the assumed criteria: the lower the value for a given criterion, the higher the ranking position, indicating a better fit according to that criterion. Then, based on the three rankings and on weights assigned to the individual criteria, an aggregate criterion (referred to as DESV) was implemented to select the best-fitting probability distribution. The method assumes that the lowest DESV value corresponds to the best-fitting theoretical distribution. In the case of the examined component, this was found to be the generalised gamma distribution. It is shown that if the final decision is based on the aggregate criterion, which takes into account the three criteria for goodness of fit, the reliability of the estimation of the time-to-failure distribution increases, and thus mistakes resulting from the use of only one of the criteria can be avoided.
Highlights
In traditional methods of estimating the parameters of the time-tofailure distribution of a technical object or its components, a specific distribution class is assumed a priori
The purpose of this article is to present the results of a procedure to identify the best-fitting probability distribution model for the time to failure of a renewable technical object using an aggregate criterion
Among the applied goodness-of-fit criteria, a particular role is played by the modified Kolmogorov–Smirnov statistic (AVGOF, average goodness of fit), which evaluates the statistical difference between the values of the empirical and theoretical CDFs
Summary
In traditional methods of estimating the parameters of the time-tofailure distribution of a technical object or its components, a specific distribution class is assumed a priori. Instead of the traditional single-criterion selection of the bestfitting family of probability distributions, the authors propose to use an aggregate criterion that includes three measures of the fit of theoretical distributions This criterion takes into account a ranking of the fit of individual probabilistic models to the empirical data, including right-censored operational data for the vehicle fleet. According to this scheme, in the first step, based on the obtained data and analysing the length of the observation time (right-censored), the survival function parameters were estimated with the Kaplan– Meier estimator and an empirical CDF was determined [7]. The times to failure of the examined component are expressed in terms of kilometres travelled, as in the paper [2]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
