Brazilian Journal of Probability and Statistics

The exponentiated logarithmic generated family of distributions and the evaluation of the confidence intervals by percentile bootstrap

Pedro Rafael Diniz Marinho, Gauss M. Cordeiro, Fernando Peña Ramírez, Morad Alizadeh, and Marcelo Bourguignon

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We study some mathematical properties of a new generator of continuous distributions with three additional parameters, called the exponentiated logarithmic generated family, to extend the normal, Weibull, gamma and Gumbel distributions, among other well-known models. Some special models are discussed. Many properties of this family are studied, some inference procedures developed and a simulation study performed to verify the adequacy of the estimators of the model parameters. We prove empirically the potentiality of the new family by means of two real data sets. The simulation study for different samples sizes assesses the performance of the maximum likelihood estimates obtained by the Swarm Optimization method. We also evaluate the performance of single and dual bootstrap methods in constructing interval estimates for the parameters. Because of the intensive simulations, we use parallel computing on a supercomputer.

Article information

Braz. J. Probab. Stat., Volume 32, Number 2 (2018), 281-308.

Received: May 2016
Accepted: November 2016
First available in Project Euclid: 17 April 2018

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Zentralblatt MATH identifier

Bootstrap generalized distribution lifetime logarithmic distribution mixture


Marinho, Pedro Rafael Diniz; Cordeiro, Gauss M.; Peña Ramírez, Fernando; Alizadeh, Morad; Bourguignon, Marcelo. The exponentiated logarithmic generated family of distributions and the evaluation of the confidence intervals by percentile bootstrap. Braz. J. Probab. Stat. 32 (2018), no. 2, 281--308. doi:10.1214/16-BJPS343.

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  • Aarset, M. V. (1987). How to identify bathtub hazard rate. IEEE Transactions on Reliability 36, 106–108.
  • Belisle, C. J. P. (1992). Convergence theorems for a class of simulated annealing algorithms on $\mathbb{R}^{d}$. Journal of Applied Probability 29, 885–895.
  • Bezanson, J., Karpinski, S., Shah, V. B. and Edelman, A. (2012). Julia: A fast dynamic language for technical computing. Preprint. Available at arXiv:1209.5145.
  • Castellares, F. and Lemonte, A. J. (2015). A new generalized Weibull distribution generated by gamma random variables. Journal of the Egyptian Mathematical Society 23, 382–390.
  • Chen, Z. (2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics and Probability Letters 49, 155–161.
  • 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.
  • Davison, A. and Hinkley, D. (1997). Bootstrap Methods and Their Application. Cambridge: Cambridge University Press.
  • Eberhart, R. C. and Kennedy, J. (1995). A new optimizer using particle swarm theory. In Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Vol. 1, 39–43.
  • Efron, B. (1979). Bootstrap methods: Another look at the Jackknife. The Annals of Statistics 7, 1–26.
  • Eugene, N., Lee, C. and Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics—Theory and Methods 31, 497–512.
  • Flajonet, P. and Odlyzko, A. (1990). Singularity analysis of generating function. SIAM Journal on Discrete Mathematics 3, 216–240.
  • Flajonet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge: Cambridge University Press.
  • Heppner, F. and Grenander, U. (1990). A stochastic nonlinear model for coordinated bird flocks. In The Ubiquity of Chaos, 233–238.
  • Lemonte, J. A. (2013). A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function. Computational Statistics and Data Analysis 62, 149–170.
  • Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability 42, 299–302.
  • Nadarajah, S. and Haghighi, F. (2011). An extension of the exponential distribution. Statistics 45, 543–558.
  • Nelder, J. A. and Mead, R. (1965). A simplex algorithm for function minimization. Computer Journal 7, 308–313.
  • Ramos, M. W. A., Marinho, P. R. D., Cordeiro, G. M., da Silva, R. V. and Hamedani, G. G. (2015). The Kumaraswamy-G Poisson family of distributions. Journal of Statistical Theory and Applications 14, 222–239.
  • Reynolds, C. W. (1987). Flocks, herds and schools: A distributed behavioral model. ACM Siggraph Computer Graphics 21, 25–34.
  • Roman, S. (1984). The Umbral Calculus, 59–63. New York: Academic Press.
  • Smith, R. L. and Naylor, J. C. (1987). A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. Applied Statistics 36, 358–369.
  • Ward, M. (1934). The representation of Stirling’s numbers and Stirling’s polynomials as sums of factorial. American Journal of Mathematics 56, 87–95.
  • Xie, M., Tang, Y. and Goh, T. N. (2002). A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering and System Safety 76, 279–285.
  • Zografos, K. and Balakrishnan, N. (2009). On families of beta- and generalized gamma-generated distributions and associated inference. Statistical Methodology 6, 344–362.