Variational Bayesian Inference for Exponentiated Weibull Right Censored Survival Data

  • Jibril Abubakar Department of Mathematics and Statistics, Faculty of Applied Science and Technology,Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Pagoh, Malaysia.
  • Mohd Asrul Affendi Abdullah Department of Mathematics and Statistics, Faculty of Applied Science and Technology,Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Pagoh, Malaysia.
  • Oyebayo Ridwan Olaniran University of Ilorin
Keywords: Exponentiated Weibull Distribution, Survival Analysis, Accelerated Failure Time, Bayesian, Variational Approximation


The exponential, Weibull, log-logistic and lognormal distributions represent the class of light and heavy-tailed distributions that are often used in modelling time-to-event data. The exponential distribution is often applied if the hazard is constant, while the log-logistic and lognormal distributions are mainly used for modelling unimodal hazard functions. The Weibull distribution is on the other hand well-known for modelling monotonic hazard rates. Recently, in practice, survival data often exhibit both monotone and non-monotone hazards. This gap has necessitated the introduction of Exponentiated Weibull Distribution (EWD) that can accommodate both monotonic and non-monotonic hazard functions. It also has the strength of adapting unimodal functions with bathtub shape. Estimating the parameter of EWD distribution poses another problem as the flexibility calls for the introduction of an additional parameter. Parameter estimation using the maximum likelihood approach has no closed-form solution, and thus, approximation techniques such as Newton-Raphson is often used. Therefore, in this paper, we introduce another estimation technique called Variational Bayesian (VB) approach. We considered the case of the accelerated failure time (AFT) regression model with covariates. The AFT model was developed using two comparative studies based on real-life and simulated data sets. The results from the experiments reveal that the Variational Bayesian (VB) approach is better than the competing Metropolis-Hasting Algorithm and the reference maximum likelihood estimates.


J. D. Kalbfleisch and R. L. Prentice, The statistical analysis of failure time data, vol. 360. JohnWiley & Sons,

O. R. Olaniran and M. A. A. Abdullah, “Bayesian analysis of extended cox model with time-varying

covariates using bootstrap prior,” Journal of Modern Applied Statistical Methods, vol. 18, no. 2, p. 7, 2019.

S. A. M. Jamil, M. A. A. Abdullah, S. L. Kek, O. R. Olaniran, and S. E. Amran, “Simulation of parametric

model towards the fixed covariate of right censored lung cancer data,” in Journal of Physics: Conference

Series, vol. 890, p. 012172, IOP Publishing, 2017.

J. Lawless, “Event history analysis and longitudinal surveys,” Analysis of Survey data, pp. 221–243, 2003.

J. Popoola, O. Popoola, and O. R. Olaniran, “An approximate performance of self-similar lognormal m 1 k

internet traffic model,” Journal of Science and Technology, vol. 11, no. 2, pp. 36–42, 2019.

J. Popoola, W. B. Yahya, O. Popoola, and O. R. Olaniran, “Generalized self-similar first order autoregressive generator (gsfo-arg) for internet traffic,” Statistics, Optimization & Information Computing, vol. 8, no. 4, pp. 810–821, 2020.

D. Ghinolfi, J. Marti, P. De Simone, Q. Lai, D. Pezzati, L. Coletti, D. Tartaglia, G. Catalano, G. Tincani, P. Carrai, et al., “Use of octogenarian donors for liver transplantation: a survival analysis,” American Journal of Transplantation, vol. 14, no. 9, pp. 2062–2071, 2014.

G. S. Mudholkar, D. K. Srivastava, and M. Freimer, “The exponentiated weibull family: A reanalysis of the bus-motor-failure data,” Technometrics, vol. 37, no. 4, pp. 436–445, 1995.

E.W. Stacy et al., “A generalization of the gamma distribution,” The Annals of mathematical statistics, vol. 33, no. 3, pp. 1187–1192, 1962.

C. Cox and M. Matheson, “A comparison of the generalized gamma and exponentiated weibull distributions,” Statistics in medicine, vol. 33, no. 21, pp. 3772–3780, 2014.

A. Pewsey, H. W. G´omez, and H. Bolfarine, “Likelihood-based inference for power distributions,” Test, vol. 21, no. 4, pp. 775–789, 2012.

V. Cancho, H. Bolfarine, and J. Achcar, “A bayesian analysis for the exponentiated-weibull distribution,” Journal of Applied Statistical Science, vol. 8, no. 4, pp. 227–242, 1999.

V. G. Cancho, J. Rodrigues, and M. de Castro, “A flexible model for survival data with a cure rate: a bayesian approach,” Journal of Applied Statistics, vol. 38, no. 1, pp. 57–70, 2011.

S. A. Khan, “Exponentiated weibull regression for time-to-event data,” Lifetime data analysis, vol. 24, no. 2, pp. 328–354, 2018.

D. M. Blei, A. Kucukelbir, and J. D. McAuliffe, “Variational inference: A review for statisticians,” Journal of the American statistical Association, vol. 112, no. 518, pp. 859–877, 2017.

O. R. Olaniran and M. A. A. Abdullah, “Subset selection in high-dimensional genomic data using hybrid variational bayes and bootstrap priors,” in Journal of Physics: Conference Series, vol. 1489, p. 012030, IOP Publishing, 2020.

S. Pasari and O. Dikshit, “Earthquake interevent time distribution in kachchh, northwestern india,” Earth, Planets and Space, vol. 67, no. 1, p. 129, 2015.

S. Pasari and O. Dikshit, “Stochastic earthquake interevent time modeling from exponentiated weibull distributions,” Natural hazards, vol. 90, no. 2, pp. 823–842, 2018.

G. S. Mudholkar and D. K. Srivastava, “Exponentiated weibull family for analyzing bathtub failure-rate data,” IEEE transactions on reliability, vol. 42, no. 2, pp. 299–302, 1993.

S. Nadarajah, “Bathtub-shaped failure rate functions,” Quality & Quantity, vol. 43, no. 5, pp. 855–863, 2009.

M. Bebbington, C.-D. Lai, and R. Zitikis, “A flexible weibull extension,” Reliability Engineering & System Safety, vol. 92, no. 6, pp. 719–726, 2007.

W. Yahya, O. Olaniran, and S. Ige, “On bayesian conjugate normal linear regression and ordinary least square regression methods: A monte carlo study,” Ilorin Journal of Science, vol. 1, no. 1, pp. 216–227, 2014.

O. Olaniran, S. Olaniran, W. Yahya, A. Banjoko, M. Garba, L. Amusa, and N. Gatta, “Improved bayesian feature selection and classification methods using bootstrap prior techniques,” Annals. Computer Science Series, vol. 14, no. 2, 2016.

O. R. Olaniran and M. A. A. B. Abdullah, “Gene selection for colon cancer classification using bayesian model averaging of linear and quadratic discriminants,” Journal of Science and Technology: Special Issue on the Application of Science and Mathematics, vol. 9, no. 3, pp. 140–144, 2017.

O. R. Olaniran and M. A. A. B. Abdullah, “Bayesrandomforest: An r implementation of bayesian random forest for regression analysis of high-dimensional data,” in Proceedings of the Third International Conference on Computing, Mathematics and Statistics (iCMS2017), pp. 269–275, Springer, 2019.

O. R. Olaniran and W. B. Yahya, “Bayesian hypothesis testing of two normal samples using bootstrap prior technique,” Journal of Modern Applied Statistical Methods, vol. 16, no. 2, p. 34, 2017.

N. M. Bala and S. bin Safei, “A hybrid harmony search and particle swarm optimization algorithm (hspso) for testing non-functional properties in software system,” Statistics, Optimization & Information Computing, 2021.

R. L. Prentice, “Exponential survivals with censoring and explanatory variables,” Biometrika, vol. 60, no. 2, pp. 279–288, 1973.

How to Cite
Abubakar, J., Abdullah, M. A. A., & Olaniran, O. R. (2023). Variational Bayesian Inference for Exponentiated Weibull Right Censored Survival Data. Statistics, Optimization & Information Computing, 11(4), 1027-1040.
Research Articles