On the parameter estimation problem of the q-Exponential distribution for reliability applications

Abstract

This work involved the q-Exponential distribution, which can be used to model each of the three phases of the bathtub curve and is an alternative to the Weibull distribution. The q-Exponential has two parameters (𝑞 – shape; 𝜂 – scale) and it stems from the Tsallis’ non-extensive entropy. This model does not have the limitation of a constant hazard rate like the Exponential one, thus allowing the modeling of either system improvement (1<𝑞<2) or degradation (𝑞<1). Besides, it has more flexibility regarding the decay of the Probability Density Function (PDF) curve and it can model very well data sets with extreme values (power law characteristic). This feature is interesting in the reliability context because many equipment can work for long time until the first failure. However, when data sets are related to the degradation phase of systems, the application of the q-Exponential distribution becomes difficult due to convergence problems in the estimation process via the maximum likelihood (ML) method. This difficulty is due to the monotone behavior of the q-Exponential log-likelihood function when 𝑞<1, which is generally known as “monotone likelihood problem”. Because of that, it is almost impossible to obtain good estimates for the parameters considering the original log-likelihood function. In this sense, this research applied the Firth’s penalization method to solve this problem. We also verified that one of the regularity conditions imposed by the ML method is not satisfied by the q-Exponential distribution. Then, with the objective of satisfying this condition, it was also proposed a variable change, which partially solved just the problems of this distribution. Nevertheless, the Firth’s method yielded satisfactory results even for small samples. Comparisons of the results were performed via Monte Carlo simulations for the original and penalized q-Exponential distribution. Additionally, bootstrap confidence intervals were constructed for the parameters and comparisons were made between the fit provided by the q-Exponential and Weibull distributions. Application examples involving failure data of complex equipment using the Firth’s penalization method are presented and discussed. The obtained results indicate that the penalized log-likelihood enables the use of the q-Exponential distribution in the modeling of data sets related to degrading systems.