An aggregate criterion for selecting a distribution

What is it about?

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.

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