Brazilian Journal of Probability and Statistics

A new lifetime model with variable shapes for the hazard rate

Abstract

We define and study a new generalization of the complementary Weibull geometric distribution introduced by Tojeiro et al. (J. Stat. Comput. Simul. 84 (2014) 1345–1362). The new lifetime model is referred to as the Kumaraswamy complementary Weibull geometric distribution and includes twenty three special models. Its hazard rate function can be constant, increasing, decreasing, bathtub and unimodal shaped. Some of its mathematical properties, including explicit expressions for the ordinary and incomplete moments, generating and quantile functions, Rényi entropy, mean residual life and mean inactivity time are derived. The method of maximum likelihood is used for estimating the model parameters. We provide some simulation results to assess the performance of the proposed model. Two applications to real data sets show the flexibility of the new model compared with some nested and non-nested models.

Article information

Source
Braz. J. Probab. Stat., Volume 31, Number 3 (2017), 516-541.

Dates
Accepted: May 2016
First available in Project Euclid: 22 August 2017

https://projecteuclid.org/euclid.bjps/1503388827

Digital Object Identifier
doi:10.1214/16-BJPS322

Mathematical Reviews number (MathSciNet)
MR3693979

Zentralblatt MATH identifier
1377.62189

Citation

Afify, Ahmed Z.; Cordeiro, Gauss M.; Shafique Butt, Nadeem; Ortega, Edwin M. M.; Suzuki, Adriano K. A new lifetime model with variable shapes for the hazard rate. Braz. J. Probab. Stat. 31 (2017), no. 3, 516--541. doi:10.1214/16-BJPS322. https://projecteuclid.org/euclid.bjps/1503388827

References

• Adamidis, K. and Loukas, S. (1998). A lifetime distribution with decreasing failure rate. Statistics and Probability Letters 39, 35–42.
• Afify, A. Z., Nofal, Z. M. and Butt, N. S. (2014). Transmuted complementary Weibull geometric distribution. Pakistan Journal of Statistics and Operation Research 10, 435–454.
• Afify, A. Z., Nofal, Z. M. and Ebraheim, A. N. (2015). Exponentiated transmuted generalized Rayleigh distribution: A new four parameter Rayleigh distribution. Pakistan Journal of Statistics and Operation Research 11, 115–134.
• Afify, A. Z., Cordeiro, G. M., Yousof, H. M., Alzaatreh, A. and Nofal, Z. M. (2016). The Kumaraswamy transmuted-$G$ family of distributions: Properties and applications. Journal of Data Science 14, 245–270.
• Barreto-Souza, W., de Morais, A. L. and Cordeiro, G. M. (2011). The Weibull-geometric distribution. Journal of Statistical Computation and Simulation 81, 645–657.
• Chen, G. and Balakrishnan, N. (1995). A general purpose approximate goodness-of-fit test. Journal of Quality Technology 27, 154–161.
• Cordeiro, G. M. and de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation 81, 883–898.
• Cordeiro, G. M., Ortega, E. M. M. and Nadarajah, S. (2010). The Kumaraswamy Weibull distribution with application to failure data. Journal of The Franklin Institute 347, 1399–1429.
• Cordeiro, G. M., Nadarajah, S. and Ortega, E. M. M. (2013). General results for the beta Weibull distribution. Journal of Statistical Computation and Simulation 83, 1082–1114.
• Cordeiro, G. M., Hashimoto, E. M. and Ortega, E. M. M. (2014). The McDonald Weibull model. Statistics: A Journal of Theoretical and Applied Statistics 48, 256–278.
• Eugene, N., Lee, C. and Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics—Theory and Methods 31, 497–512.
• Gomes, A. E., da-Silva, C. Q., Cordeiro, G. M. and Ortega, E. M. M. (2014). A new lifetime model: The Kumaraswamy generalized Rayleigh distribution. Journal of Statistical Computation and Simulation 84, 290–309.
• Guess, F. and Proschan, F. (1988). Mean residual life, theory and applications. Handbook of Statistics 7, 215–224.
• Gupta, R. D. and Kundu, D. (1999). Generalized exponential distributions: Theory and methods. Australian and New Zealand Journal of Statistics 41, 173–188.
• Gupta, R. D. and Kundu, D. (2001). Exponentiated exponential family: An alternative to gamma and Weibull distributions. Biometrical Journal 43, 117–130.
• Khan, M. N. (2015). The modified beta Weibull distribution. Hacettepe Journal of Mathematics and Statistics 44, 1553–1568.
• Kundu, D. and Raqab, M. Z. (2005). Generalized Rayleigh distribution: Different methods of estimations. Computational Statistics and Data Analysis 49, 187–200.
• Kundu, D. and Raqab, M. Z. (2009). Estimation of $R=P(Y<X)$ for three parameter Weibull distribution. Statistics and Probability Letters 79, 1839–1846.
• Lai, C. D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer Science and Business Media. Berlin: Springer.
• Lee, C., Famoye, F. and Olumolade, O. (2007). Beta-Weibull distribution: Some properties and applications to censored data. Journal of Modern Applied Statistical Methods 6, 173–186.
• Louzada, F., Roman, M. and Cancho, V. G. (2011). The complementary exponential geometric distribution: Model, properties, and a comparison with its counterpart. Computational Statistics and Data Analysis 55, 2516–2524.
• Louzada, F., Marchi, V. and Carpenter, J. (2013). The complementary exponentiated exponential geometric lifetime distribution. Journal of Probability and Statistics 2013, Article ID 502159.
• Mudholkar, G. S., Srivastava, D. K. and Kollia, G. D. (1996). A generalization of the Weibull distribution with application to the analysis of survival data. Journal of the American Statistical Association 91, 1575–1583.
• Nadarajah, S., Cordeiro, G. M. and Ortega, E. M. M. (2013). The exponentiated Weibull distribution: A survey. Statistical Papers 54, 839–877.
• Nassar, M. M. and Eissa, F. H. (2003). On the exponentiated Weibull distribution. Communications in Statistics—Theory and Methods 32, 1317–1336.
• Nofal, Z. M., Afify, A. Z., Yousof, H. M. and Cordeiro, G. M. (2017). The generalized transmuted-$G$ family of distributions. Communications in Statistics—Theory and Methods 46, 4119–4136.
• Perdoná, G. S. C. (2006). Modelos de Riscos Aplicados a Análise de Sobrevivência. Doctoral thesis, Institute of Computer Science and Mathematics, University of São Paulo, Brasil (in Portuguese).
• Rayleigh, J. W. S. (1880). On the resultant of a large number of vibration of the same pitch and arbitrary phase. Philosophical Magazine, 5th Series 10, 73–78.
• Rinne, H. (2009). The Weibull Distribution: A Handbook. London: CRC Press.
• Silva, A. N. F. (2004). Estudo evolutivo das crianças expostas ao HIV e notificadas pelo núcleo de vigilância epidemiolôgica do HCFMRP-USP. M.Sc. thesis, University of São. Paulo, Brasil (in Portuguese).
• Tojeiro, C., Louzada, F., Roman, M. and Borges, P. (2014). The complementary Weibull geometric distribution. Journal of Statistical Computation and Simulation 84, 1345–1362.
• Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, Transactions, ASME 18, 293–297.