Improved new modified Weibull distribution: A Bayes study using Hamiltonian Monte Carlo simulation

Abstract

The newly modified Weibull distribution defined in the literature is a model based on combining the Weibull and modified Weibull distributions. It has been demonstrated as the best model for fitting to the bathtub-shaped failure rate data sets. However, another new model based on combining the modified Weibull and Gompertz distributions has been demonstrated later to be even better than the first model. In this article, we have shown how to improve the former model into a better model, and more importantly, we have provided a full Bayesian analysis of the improved model. The Hamiltonian Monte Carlo and cross-entropy methods have been exploited to empower the traditional methods of statistical estimation. Bayes estimators have been obtained using Hamiltonian Monte Carlo for posterior simulations. Bayesian model checking has also been provided in order to check the validation of the model when fitting to real data sets. We have also provided the maximum likelihood estimators of the model parameters using the cross-entropy method to optimize the log-likelihood function. The results derived from the analysis of two well-known data sets show that the improved model is much better than its original form.

An additive Chen‐Weibull distribution and its applications in reliability modeling

“…We refer the reader to Neal 35 and Gelman et al 36 for excellent explanations of this algorithm. A step‐by‐step reexplanation of the algorithm can also be found in our previous publications Thach and Bris 22 and Thach et al 23 Notice that HMC is build to work with all continuous positive target densities. In this study, we use Rstan 37 to sample from the posterior distribution.…”

“…For the details of this method, we refer the readers to Kroese at al 20 and Rubinstein and Kroese. 21 A step-by-step reexplanation of the algorithm can also be found in our previous publications Thach and Bris 22 and Thach et al 23 For R users, we recommend using the package "CEoptim." 24 Using the invariance property of MLE, the MLEs of R(t) and h(t) are obtained bŷ…”

Classical and Bayesian estimations of improved Weibull–Weibull distribution for complete and censored failure times data

“…This can be interpreted as the model not being flexible enough to accommodate bathtub-shaped FR data sets well. To modify the model, we adopt a similar technique by Thach and Bris 33 and reparameterized the component exp(𝜃x 𝜏 ) to be always increasing (by limiting 𝜏 > 1). Thus, we have the new expression for the FRF given as…”

A Three-Component Additive Weibull Distribution and Its Reliability Implications

“…In our previous studies [6,11] we realized that if we combine a model which has a bathtub-shaped failure rate function with a model which has an increasing failure rate function, we might obtain a model which has a very flexible bathtub-shaped failure rate function with a long flat region. In fact, we encountered such distribution in the literature, such as [4,7,10,11].…”

“…It has five parameters and its CDF is given by: F(x) = 1 − e −(αx θ +βx γ e λx ) , x ≥ 0, where α, β, γ, θ > 0, and λ ≥ 0. A Bayes study of the improved NMW by using Hamiltonian Monte Carlo simulation is given in [6]. The additive modified Weibull (AMW) distribution [7] combines the modified Weibull distribution and the Gompertz distribution [8].…”