Brazilian Journal of Probability and Statistics

The complementary exponential power series distribution

José Flores D., Patrick Borges, Vicente G. Cancho, and Francisco Louzada

Full-text: Open access


In this paper, we introduce the complementary exponential power series distributions, with failure rate increasing, which is complementary to the exponential power series model proposed by Chahkandi and Ganjali [Comput. Statist. Data Anal. 53 (2009) 4433–4440]. The new class of distribution arises on latent complementary risks scenarios, where the lifetime associated with a particular risk is not observable, rather we observe only the maximum lifetime value among all risks. This new class contains several distributions as a particular case. The properties of the proposed distribution class are discussed, such as quantiles, moments and order statistics. Estimation is carried out via maximum likelihood. Simulation results on maximum likelihood estimation are presented. A real application illustrates the usefulness of the new distribution class.

Article information

Braz. J. Probab. Stat., Volume 27, Number 4 (2013), 565-584.

First available in Project Euclid: 9 September 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Complementary risks exponential distribution increasing failure rate power series distribution exponential power series distribution


Flores D., José; Borges, Patrick; Cancho, Vicente G.; Louzada, Francisco. The complementary exponential power series distribution. Braz. J. Probab. Stat. 27 (2013), no. 4, 565--584. doi:10.1214/11-BJPS182.

Export citation


  • Aarset, M. V. (1985). The null distribution for a test of constant versus “bathtub” failure rate. Scandinavian Journal of Statistics 12(1), 55–68.
  • Adamidis, K. and Loukas, S. (1998). A lifetime distribution with decreasing failure rate. Statistics & Probability Letters 39(1), 35–42.
  • Adamidis, K., Dimitrakopoulou, T. and Loukas, S. (2005). On an extension of the exponential-geometric distribution. Statistics & Probability Letters 73(3), 259–269.
  • Barakat, H. M. and Abdelkader, Y. H. (2004). Computing the moments of order statistics from nonidentical random variables. Statistical Methods and Applications 13, 15–26.
  • Barreto-Souza, W. and Cribari-Neto, F. (2009). A generalization of the exponential–Poisson distribution. Statistics & Probability Letters 79, 2493–2500.
  • Basu, A. and Klein, J. (1982). Some recent development in competing risks theory. In Survival Analysis (Crowley, J. and Johnson, R. A, eds.) 1, 216–229. Hayward: IMS.
  • Cancho, V. G., Louzada-Neto, F. and Barriga, G. D. (2011). The Poisson–exponential lifetime distribution. Computational Statistics & Data Analysis 55, 677–686.
  • Chahkandi, M. and Ganjali, M. (2009). On some lifetime distributions with decreasing failure rate. Computational Statistics & Data Analysis 53, 4433–4440.
  • Cordeiro, G., Rodrigues, J. and de Castro, M. (2012). The exponential COM–Poisson distribution. Statistical Papers 53, 653–664.
  • Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics. London: Chapman & Hall.
  • Cox, D. R. and Oakes, D. (1984). Analysis of Survival Data. London: Chapman & Hall.
  • Crowder, M., Kimber, A., Smith, R. and Sweeting, T. (1991). Statistical Analysis of Reliability Data. London: Chapman & Hall.
  • Goetghebeur, E. and Ryan, L. (1995). A modified log rank test for competing risks with missing failure type. Biometrics 77, 207–211.
  • Johnson, N. L., Kemp, A. W. and Kotz, S. (2005). Univariate Discrete Distribution. Hoboken, NJ: Wiley.
  • Kus, C. (2007). A new lifetime distribution. Computational Statistics & Data Analysis 51, 4497–4509.
  • Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data, 2nd ed. Hoboken, NJ: Wiley.
  • Lu, K. and Tsiatis, A. A. (2001). Multiple imputation methods for estimating regression coefficients in the competing risks model with missing cause of failure. Biometrics 54, 1191–1197.
  • Lu, K. and Tsiatis, A. A. (2005). Comparision between two partial likelihood approaches for the competing risks model with missing cause of failure. Lifetime Data Analysis 11, 29–40.
  • Morais, A. L. and Barreto-Souza, W. (2011). A compound class of weibull and power series distributions. Computational Statistics & Data Analysis 55(3), 1410–1425.
  • Reiser, B., Guttman, I., Lin, D., Guess, M. and Usher, J. (1995). Bayesian inference for masked system lifetime data. Applied Statistics 44, 79–90.
  • R Development Core Team (2008). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
  • Silva, R. B., Barreto-Souza, W. and Cordeiro, G. M. (2010). A new distribution with decreasing, increasing and upside-down bathtub failure rate. Computational Statistics & Data Analysis 54(4), 935–944.
  • Tahmashi, R. and Rezaei, S. (2008). A two-parameter lifetime distribution with decreasing failure rate. Computational Statistics & Data Analysis 52, 3889–3901.