Cross-comparative Analysis of Frequentist and Bayesian Perspectives on Failure Time Distributions

Abstract

In reliability analysis, both the Weibull and the lognormal distributions could be analyzed using the observed data logarithms. While the Weibull data logarithm is skewed, the lognormal data logarithm is symmetrical. This review work initiates discussions on and syntheses of the various views held on the use of the frequentist and Bayesian approaches to drawing statistical inferences on failure time distributions of survival models. Of greater concern was the discussion on the use of the exponential, Weibull and log-normal distributions in reliability analysis. Various methods have been used to discriminate between the two most important distributions. They include: Coefficients of variation (CV); the standard deviation of the data logarithms (SD); the percentile position of the mean of the data logarithm (PP); the cumulated logarithm dispersion before and after the mean (CLD); ratio of the maximum likelihood (RML); Kullback-Leibler Divergence (KLD); and Minimized Kullback-Leibler Divergence (RMKLD). In the (CV, SD, PP and CLD) study, a stress-strength data set was used for the analysis. The stress data followed a lognormal distribution, while the strength data followed a Weibull distribution, therefore for the stress-strength analysis the lognormal-Weibull combination was used. In the ratio of the maximum likelihood (RML) study, it was averred that though each of the distributions had great applications, none of them produced a good fit. In the study using Kullback-Leibler Divergence (KLD); and Minimized Kullback-Leibler Divergence (RMKLD), test results revealed that RML=0.345>0. Hence the lognormal distribution was selected. Similarly, the RMKLD=0.6028>0, therefore the Weibull distribution was selected. In the final analysis, none of the study results could reject the Weibull in favour of the lognormal distribution model. In respect of the frequentist and the Bayesian approach to conducting statistical inferences, it came out strongly that it was high time psychologist who had adopted the use of the frequentist framework moved away from it and started using the Bayesian approach. A shift towards the use of the Bayesian, lognormal-Weibull approach is thus recommendable.