Brazilian Journal of Probability and Statistics

Log-symmetric distributions: Statistical properties and parameter estimation

Luis Hernando Vanegas and Gilberto A. Paula

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In this paper, we study the main statistical properties of the class of log-symmetric distributions, which includes as special cases bimodal distributions as well as distributions that have heavier/lighter tails than those of the log-normal distribution. This family includes distributions such as the log-normal, log-Student-$t$, harmonic law, Birnbaum–Saunders, Birnbaum–Saunders-$t$ and generalized Birnbaum–Saunders. We derive quantile-based measures of location, dispersion, skewness, relative dispersion and kurtosis for the log-symmetric class that are appropriate in the context of asymmetric and heavy-tailed distributions. Additionally, we discuss parameter estimation based on both classical and Bayesian approaches. The usefulness of the log-symmetric class is illustrated through a statistical analysis of a real dataset, in which the performance of the log-symmetric class is compared with that of some competitive and very flexible distributions.

Article information

Braz. J. Probab. Stat., Volume 30, Number 2 (2016), 196-220.

Received: March 2014
Accepted: November 2014
First available in Project Euclid: 31 March 2016

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Zentralblatt MATH identifier

Skewness asymmetric distributions Bayesian analysis robust estimates log-location-scale lifetime models


Vanegas, Luis Hernando; Paula, Gilberto A. Log-symmetric distributions: Statistical properties and parameter estimation. Braz. J. Probab. Stat. 30 (2016), no. 2, 196--220. doi:10.1214/14-BJPS272.

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  • Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In International Symposium on Information Theory (B. N. Petrov and F. Csaki, eds.) 267–281. Budapest, Hungary: Akademiai Kiado.
  • Andrews, D. R. and Mallows, C. L. (1974). Scale mixtures of normal distributions. Journal of the Royal Statistical Society. Series B 36, 99–102.
  • Azzalini, A., dal Cappello, T. and Kotz, S. (2003). Log-skew-normal and log-skew-$t$ distributions as model for family income data. Journal of Income Distribution 11, 12–20.
  • Balakrishnan, N., Leiva, V., Sanhueza, A. and Vilca, F. (2009). Estimation in the Birnbaum–Saunders distributions based on scale-mixture of normals and the EM-algorithm. Statistics and Operations Research Transactions (SORT) 33, 171–192.
  • Barndoff-Nielsen, O. (1977). Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 353, 401–419.
  • Barros, M., Paula, G. A. and Leiva, V. (2008). A new class of survival regression models with heavy-tailed errors: Robustness and diagnostics. Lifetime Data Analysis 14, 316–332.
  • Birnbaum, Z. W. and Saunders, S. C. (1969). A new family of life distributions. Journal of Applied Probability 6, 319–327.
  • Bonett, D. G. (2006). Confidence interval for a coefficient of quartile variation. Computational Statistics & Data Analysis 50, 2953–2957.
  • Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Reading, MA: Addison-Wesely Publishing Company.
  • Carrasco, J. M. F., Ortega, E. M. M. and Cordeiro, G. M. (2008). A generalized modified Weibull distribution for lifetime modeling. Computational Statistics & Data Analysis 53, 450–462.
  • Chib, S. and Greenberg, E. (1995). Understanding the Metropolis–Hastings algorithm. The American Statistician 49, 327–335.
  • Cordeiro, G. M., Ferrari, S. L. P., Uribe-Opazo, M. A. and Vasconcellos, K. L. P. (2000). Corrected maximum-likelihood estimation in a class of symmetric nonlinear regression models. Statistics & Probability Letters 46, 317–328.
  • Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B 39, 1–38.
  • Díaz-García, J. A. and Leiva, V. (2005). A new family of life distributions based on elliptically contoured distributions. Journal of Statistical Planning and Inference 128, 445–457.
  • Fang, K. T., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions. London: Chapman & Hall.
  • Fonseca, T. C. O., Migon Helio, S. and Ferreira Marco, A. R. (2012). Bayesian analysis based on the Jeffreys prior for the hyperbolic distribution. Brazilian Journal of Probability and Statistics 26, 327–343.
  • Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association 85, 398–409.
  • Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (2004). Bayesian Data Analysis. Boca Raton: Chapman & Hall.
  • Glaser, R. E. (1980). Bathtub and related failure rate characterizations. Journal of the American Statistical Association 75, 667–672.
  • Groeneveld, R. A. and Meeden, G. (1984). Measuring skewness and kurtosis. The Statistician 33, 391–399.
  • Hinkley, D. V. (1975). On power transformations to symmetry. Biometrika 62, 101–111.
  • Hörmann, W. and Leydold, J. (2015). Generating generalized inverse Gaussian random variates. Statistics and Computing 24, 547–557.
  • Jørgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. New York: Springer.
  • Kano, Y., Berkane, M. and Bentler, P. (1993). Statistical inference based on pseudo-maximum likelihood estimators in elliptical populations. Journal of the American Statistical Association 88, 135–143.
  • Leiva, V., Riquelme, M., Balakrishnan, N. and Sanhueza, A. (2008). Lifetime analysis based on the generalized Birnbaum–Saunders distribution. Computational Statistics & Data Analysis 52, 2079–2097.
  • Limpert, E., Stahel, W. A. and Abbt, M. (2001). Log-normal distributions across the sciences: Key and clues. BioScience 51, 341–352.
  • Lucas, A. (1997). Robustness of the Student-$t$ based $M$-estimator. Communications in Statistics, Theory and Methods 26, 1165–1182.
  • Marchenko, Y. V. and Genton, M. G. (2010). Multivariate log-skew-elliptical distributions with applications to precipitation data. Environmetrics 21, 318–340.
  • Marshall, A. W. and Olkin, I. (2007). Life Distributions. New York: Springer.
  • Moors, J. J. A. (1988). A quantile alternative for kurtosis. The Statistician 37, 25–32.
  • Nadarajah, S. and Kotz, S. (2006). The exponentiated type distributions. Acta Applicandae Mathematicae 92, 97–111.
  • Paula, G. A., Leiva, V., Barros, M. and Liu, S. (2012). Robust statistical modeling using the Birnbaum–Saunders-$t$ distribution applied to insurance distribution. Applied Stochastic Model in Business and Industry 28, 16–34.
  • Puig, P. (2008). A note on the harmonic law: A two-parameter family of distributions for ratios. Statistics & Probability Letters 78, 320–326.
  • R Core Team (2013). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. Available at
  • Rieck, J. R. (1999). A moment-generating function with application to the Birnbaum–Saunders distribution. Communications in Statistics—Theory and Methods 28, 2213–2222.
  • Rieck, J. R. and Nedelman, J. R. (1991). A log-linear model for the Birnbaum–Saunders distribution. Technometrics 33, 51–60.
  • Rigby, R. A. and Stasinopoulos, D. M. (2006). Using the box-cox $t$ distribution in GAMLSS to model skewness and kurtosis. Statistical Modelling 6, 209–229.
  • Rogers, W. H. and Tukey, J. W. (1972). Understanding some long-tailed symmetrical distributions. Statistica Neerlandica 26, 211–226.
  • Schwarz, G. E. (1978). Estimating the dimension of a model. Annals of Statistics 6, 461–464.
  • Stacy, E. W. (1962). A generalization of the gamma distribution. The Annals of Mathematical Statistics 33, 1187–1192.
  • Villegas, C., Paula, G. A., Cysneiros, F. J. A. and Galea, M. (2013). Influence diagnostics in generalized symmetric linear models. Computational Statistics & Data Analysis 59, 161–170.
  • West, M. (1987). On scale mixtures of normal distributions. Biometrika 74, 646–648.
  • Zwillinger, D. and Kokoska, S. (2000). Standard Probability and Statistical Tables and Formula. Boca Raton: Chapman & Hall.