Brazilian Journal of Probability and Statistics

The beta log-logistic distribution

Artur J. Lemonte

Full-text: Open access

Abstract

A new continuous distribution, so-called the beta log-logistic distribution, that extends the log-logistic distribution and some other distributions is proposed and studied. The new model is quite flexible to analyze positive data. Various structural properties of the new distribution are derived, including explicit expressions for the moments, mean deviations and Rényi and Shannon entropies. The score function is derived and the estimation of the model parameters is performed by maximum likelihood. We also determine the expected information matrix. The usefulness of the new model is illustrated by means of two real data sets. We hope that the new distribution proposed here will serve as an alternative model to other models available in the literature for modeling positive real data in many areas.

Article information

Source
Braz. J. Probab. Stat., Volume 28, Number 3 (2014), 313-332.

Dates
First available in Project Euclid: 17 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1405603504

Digital Object Identifier
doi:10.1214/12-BJPS209

Mathematical Reviews number (MathSciNet)
MR3263050

Zentralblatt MATH identifier
1312.60008

Keywords
Entropy fisk distribution log-logistic distribution maximum likelihood estimation mean deviations moments

Citation

Lemonte, Artur J. The beta log-logistic distribution. Braz. J. Probab. Stat. 28 (2014), no. 3, 313--332. doi:10.1214/12-BJPS209. https://projecteuclid.org/euclid.bjps/1405603504


Export citation

References

  • Ahmad, M. I., Sinclair, C. D. and Werritty, A. (1988). Log-logistic flood frequency analysis. Journal of Hydrology 98, 205–224.
  • Akinsete, A., Famoye, F. and Lee, C. (2008). The beta-Pareto distribution. Statistics 42, 547–563.
  • Ashkar, F. and Mahdi, S. (2006). Fitting the log-logistic distribution by generalized moments. Journal of Hydrology 328, 694–703.
  • Barreto–Souza, W., Santos, A. H. S. and Cordeiro, G. M. (2010). The beta generalized exponential distribution. Journal of Statistical Computation and Simulation 80, 159–172.
  • Birnbaum, Z. W. and Saunders, S. C. (1969). A new family of life distributions. Journal of Applied Probability 6, 319–327.
  • 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 Brito, R. S. (2012). The beta power distribution. Brazilian Journal of Probability and Statistics 26, 88–112.
  • Cordeiro, G. M., Cristino, C. T., Hashimoto, E. M. and Ortega, E. M. M. (2013). The beta generalized Rayleigh distribution with applications to lifetime data. Statistical Papers 54, 133–161.
  • Cordeiro, G. M. and Lemonte, A. J. (2011a). The $\beta$-Birnbaum–Saunders distribution: An improved distribution for fatigue life modeling. Computational Statistics and Data Analysis 55, 1445–1461.
  • Cordeiro, G. M. and Lemonte, A. J. (2011b). The beta Laplace distribution. Statistics and Probability Letters 81, 973–982.
  • Cordeiro, G. M. and Lemonte, A. J. (2011c). The beta-half-Cauchy distribution. Journal of Probability and Statistics 2011, 1–18.
  • 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., Ortega, E. M. M. and Silva, G. O. (2012). The beta extended Weibull family. Journal of Probability and Statistical Science 10, 15–40.
  • Cordeiro, G. M., Silva, G. O. and Ortega, E. M. M. (2014). The beta-Weibull geometric distribution. Statistics 47, 817–834.
  • Dey, A. K. and Kundu, D. (2010). Discriminating between the log-normal and log-logistic distributions. Communications in Statistics—Theory and Methods 39, 280–292.
  • Doornik, J. A. (2006). An Object-Oriented Matrix Language—Ox 4, 5th ed. London: Timberlake Consultants Press.
  • Eugene, N., Lee, C. and Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics—Theory and Methods 31, 497–512.
  • Fisk, P. R. (1961). The graduation of income distributions. Econometrica 29, 171–185.
  • Garvan, F. (2002). The Maple Book. London: Chapman & Hall/CRC.
  • Ghitany, M. E., Al-Hussaini, E. K. and AlJarallah, R. A. (2005). Marshall–Olkin extended Weibull distribution and its application to censored data. Journal of Applied Statistics 32, 1025–1034.
  • Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products. New York: Academic Press.
  • Hansen, B. E. (1994). Autoregressive conditional density estimation. International Economic Review 35, 705–730.
  • Hosking, J. R. M. (1990). L-moments: Analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society, Ser. B 52, 105–124.
  • Johnson, N., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions—Volume 1, 2nd ed. New York: Wiley.
  • Johnson, N., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions—Volume 2, 2nd ed. New York: Wiley.
  • Jones, M. C. (2004). Families of distributions arising from distributions of order statistics. Test 13, 1–43.
  • Kenney, J. F. and Keeping, E. S. (1962). Mathematics of Statistics, Part 1, 3rd ed. Princeton, NJ: Van Nostrand.
  • 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.
  • Lee, E. T. and Wang, J. W. (2003). Statistical Methods for Survival Data Analysis, 3rd ed. New York: Wiley.
  • Moors, J. J. A. (1998). A quantile alternative for kurtosis. Journal of the Royal Statistical Society, Ser. D 37, 25–32.
  • 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 Gupta, A. K. (2004). The beta Fréchet distribution. Far East Journal of Theoretical Statistics 14, 15–24.
  • Nadarajah, S. and Kotz, S. (2004). The beta Gumbel distribution. Mathematical Problems in Engineering 10, 323–332.
  • Nadarajah, S. and Kotz, S. (2006). The beta exponential distribution. Reliability Engineering and System Safety 91, 689–697.
  • Nichols, M. D. and Padgett, W. J. (2006). A Bootstrap control chart for Weibull percentiles. Quality and Reliability Engineering International 22, 141–151.
  • Parnaíba, P. F., Ortega, E. M. M., Cordeiro, G. M. and Pescim, R. R. (2011). The beta Burr XII distribution with application to lifetime data. Computational Statistics and Data Analysis 55, 1118–1136.
  • Pescim, R. R., Demétrio, C. G. B., Cordeiro, G. M., Ortega, E. M. M. and Urbano, M. R. (2010). The beta generalized half-normal distribution. Computational Statistics and Data Analysis 54, 945–957.
  • R Development Core Team (2012). R: A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing.
  • Sigmon, K. and Davis, T. A. (2002). MATLAB Primer, 6th ed. London: Chapman & Hall/CRC.
  • Silva, G. O., Ortega, E. M. M. and Cordeiro, G. M. (2010). The beta modified Weibull distribution. Lifetime Data Analysis 16, 409–430.
  • Shoukri, M. M., Mian, I. U. M. and Tracy, D. S. (1988). Sampling properties of estimators of the log-logistic distribution with application to canadian precipitation data. Canadian Journal of Statistics 16, 223–236.
  • Wolfram, S. (2003). The Mathematica Book, 5th ed. London: Cambridge Univ. Press.