Electronic Journal of Statistics

Reparameterized Birnbaum-Saunders regression models with varying precision

Manoel Santos-Neto, Francisco José A. Cysneiros, Víctor Leiva, and Michelli Barros

Full-text: Open access


We propose a methodology based on a reparameterized Birnbaum-Saunders regression model with varying precision, which generalizes the existing works in the literature on the topic. This methodology includes the estimation of model parameters, hypothesis tests for the precision parameter, a residual analysis and influence diagnostic tools. Simulation studies are conducted to evaluate its performance. We apply it to two real-world case-studies to show its potential with the R software.

Article information

Electron. J. Statist., Volume 10, Number 2 (2016), 2825-2855.

Received: July 2014
First available in Project Euclid: 30 September 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J12: Generalized linear models 62J20: Diagnostics
Secondary: 62F03: Hypothesis testing

Birnbaum-Saunders distribution hypothesis testing likelihood-based methods local influence Monte Carlo simulation residuals R software


Santos-Neto, Manoel; Cysneiros, Francisco José A.; Leiva, Víctor; Barros, Michelli. Reparameterized Birnbaum-Saunders regression models with varying precision. Electron. J. Statist. 10 (2016), no. 2, 2825--2855. doi:10.1214/16-EJS1187. https://projecteuclid.org/euclid.ejs/1475266648

Export citation


  • [1] Bhatti, C. (2010). The Birnbaum-Saunders autoregressive conditional duration model., Mathematics and Computers in Simulation, 80 2062–2078. doi:10.1016/j.matcom.2010.01.011.
  • [2] Birnbaum, Z. W. and Saunders, S. C. (1969). A new family of life distributions., Journal of Applied Probability, 6 319–327. doi:10.2307/3212003.
  • [3] Cook, R. D. and Weisberg, S. (1983). Diagnostics for heteroscedasticity in regression., Biometrika, 70 1–10. doi:10.2307/2335938.
  • [4] Cox, D. and Hinkley, D. (1974)., Theoretical Statistics. Chapman and Hall, London, UK.
  • [5] Cysneiros, F., Paula, G., and Galea, M. (2007). Heteroscedastic symmetrical linear models., Statistical and Probability Letters, 77 1084–1090. doi:10.1016/j.spl.2007.01.012.
  • [6] Dunn, P. and Smyth, G. (1996). Randomized quantile residuals., Journal of Computational and Graphical Statistics, 5 236–244. doi:10.2307/1390802
  • [7] Ferrari, S., Espinheira, P., and Cribari-Neto, F. (2011). Diagnostic tools in beta regression with varying dispersion., Statistica Neerlandica, 65 337–351. doi:10.1111/j.1467-9574.2011.00488.x.
  • [8] Ferreira, M., Gomes, M. I., and Leiva, V. (2012). On an extreme value version of the Birnbaum-Saunders distribution., REVSTAT Statistical Journal, 10 181–210. https://www.ine.pt/revstat/pdf/rs120202.pdf.
  • [9] Galea, M., Leiva, V., and Paula, G. (2004). Influence diagnostics in log-Birnbaum-Saunders regression models., Journal of Applied Statistics, 31 1049–1064. doi:10.1080/0266476042000280409.
  • [10] Garcia-Papani, F., Uribe-Opazo, M., Leiva, V., and Aykroyd, R. (2016). Birnbaum-Saunders spatial modelling and diagnostics applied to agricultural engineering data., Stochastic Environmental Research and Risk Assessment, pages in press available at doi:10.1007/s00477-015-1204-4.
  • [11] Jin, X. and Kawczak, J. (2003). Birnbaum-Saunders and lognormal kernel estimators for modelling durations in high frequency financial data., Annals of Economics and Finance, 4 103–124. http://aeconf.com/Articles/May2003/aef040106.pdf.
  • [12] Johnson, N., Kotz, S., and Balakrishnan, N. (1995)., Continuous Univariate Distributions, volume 2. Wiley, New York, US.
  • [13] Leiva, V. (2016)., The Birnbaum-Saunders Distribution. Academic Press, New York, US.
  • [14] Leiva, V., Marchant, C., Ruggeri, F., and Saulo, H. (2015a). A criterion for environmental assessment using Birnbaum-Saunders attribute control charts., Environmetrics, 26 463–476. doi:10.1002/env.2349.
  • [15] Leiva, V., Rojas, E., Galea, M., and Sanhueza, A. (2014a). Diagnostics in Birnbaum-Saunders accelerated life models with an application to fatigue data., Applied Stochastic Models in Business and Industry, 30 115–131. doi:10.1002/asmb.1944.
  • [16] Leiva, V., Santos-Neto, M., Cysneiros, F. J. A., and Barros, M. (2014b). Birnbaum-Saunders statistical modelling: A new approach., Statistical Modelling, 14 21–48. doi:10.1177/1471082X13494532.
  • [17] Leiva, V., Santos-Neto, M., Cysneiros, F. J. A., and Barros, M. (2016). A methodology for stochastic inventory models based on a zero-adjusted Birnbaum-Saunders distribution., Applied Stochastic Models in Business and Industry, 32 74–89. doi:10.1002/asmb.2124.
  • [18] Leiva, V., Saulo, H., Leão, J., and Marchant, C. (2014c). A family of autoregressive conditional duration models applied to financial data., Computational Statistics and Data Analysis, 79 175–191. doi:10.1016/j.csda.2014.05.016.
  • [19] Leiva, V., Tejo, M., Guiraud, P., Schmachtenberg, O., Orio, P., and Marmolejo, F. (2015b). Modeling neural activity with cumulative damage distributions., Biological Cybernetics, 109 421–433. doi:10.1007/s00422-015-0651-9.
  • [20] Li, A., Chen, Z., and Xie, F. (2012). Diagnostic analysis for heterogeneous log-Birnbaum-Saunders regression models., Statistical and Probability Letters, 89 1690–1698. doi:10.1016/j.spl.2012.05.021.
  • [21] Lin, J., Zhu, L., and Xie, F. (2009). Heteroscedasticity diagnostics for $t$ linear regression models., Metrika, 70 59–77. doi:10.1007/s00184-008-0179-2.
  • [22] Marchant, C., Leiva, V., and Cysneiros, F. (2016a). A multivariate log-linear model for Birnbaum-Saunders distributions., IEEE Transactions on Reliability, 65 816–827 doi:10.1109/TR.2015.2499964.
  • [23] Marchant, C., Leiva, V., Cysneiros, F., and Vivanco, J. (2016b). Diagnostics in multivariate generalized Birnbaum-Saunders regression models., Journal of Applied Statistics, 43 2829–2849 doi:10.1080/02664763.2016.1148671.
  • [24] Owen, W. (2006). A new three-parameter extension to the Birnbaum-Saunders distribution., IEEE Transactions on Reliability, 55 475–479. doi:10.1109/TR.2006.879646.
  • [25] Owen, W. and Padgett, W. (2000). A Birnbaum-Saunderss accelerated life model., IEEE Transactions on Reliability, 49 224–229. doi:10.1109/24.877342.
  • [26] Paula, G. A. (2013). On diagnostics in double generalized linear models., Computational Statistics and Data Analysis, 68 44–51. doi:10.1016/j.csda.2013.06.008.
  • [27] Paula, G. A., Leiva, V., Barros, M., and Liu, S. (2012). Robust statistical modeling using the Birnbaum-Saunders-t distribution applied to insurance., Applied Stochastic Models in Business and Industry, 28 16–34. doi:10.1002/asmb.887.
  • [28] Qu, H. and Xie, F. (2011). Diagnostics analysis for log-Birnbaum-Saunders regression models with censored data., Statistica Neerlandica, 65 1–21. doi:10.1111/j.1467-9574.2010.00467.x.
  • [29] R-Team (2016)., R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://cran.r-project.org/doc/manuals/r-release/fullrefman.pdf.
  • [30] Rieck, J. and Nedelman, J. (1991). A log-linear model for the Birnbaum-Saunders distribution., Technometrics, 3 51–60. http://www.jstor.org/stable/1269007.
  • [31] Rocha, A. V. and Simas, A. B. (2011). Influence diagnostic in a general class of beta regression models., TEST, 20 95–119. doi:10.1007/s11749-010-0189-z.
  • [32] Rojas, F., Leiva, V., Wanke, P., and Marchant, C. (2015). Optimization of contribution margins in food services by modeling independent component demand., Revista Colombiana de Estadística, 38 1–30. doi:10.15446/rce.v38n1.48799.
  • [33] Santos-Neto, M., Cysneiros, F., Leiva, V., and Barros, M. (2016)., RBS: Reparameterized Birnbaum-Saunders regression model. R package version 0.0.1 https://github.com/santosneto/RBS.
  • [34] Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., and Ahmed, S. (2012). On new parameterizations of the Birnbaum-Saunders distribution., Pakistan Journal of Statistics, 28 1–26.
  • [35] Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., and Barros, M. (2014). On new parameterizations of the Birnbaum-Saunders distribution and its moments, estimation and application., REVSTAT Statistical Journal, 12 247–272. https://www.ine.pt/revstat/pdf/rs140303.pdf.
  • [36] Saulo, H., Leiva, V., Ziegelmann, F. A., and Marchant, C. (2013). A nonparametric method for estimating asymmetric densities based on skewed Birnbaum-Saunders distributions applied to environmental data., Stochastic Environmental Research and Risk Assessment, 27 1479–1491. doi:10.1007/s00477-012-0684-8
  • [37] Saumard, A. (2013). Optimal model selection in heteroscedastic regression using piecewise polynomial functions., Electronic Journal of Statistics, 7 1184–1223. doi:10.1214/13-EJS803.
  • [38] Simas, A. B., Barreto-Souza, W., and Rocha, A. V. (2010). Improved estimators for a general class of beta regression models., Computational Statistics and Data Analysis, 54 348–366. doi:10.1016/j.csda.2009.08.017.
  • [39] Smyth, G. (1989). Generalized linear models with varying dispersion., Journal of the Royal Statistical Society B, 51 47–60. http://www.jstor.org/stable/2345840.
  • [41] Stasinopoulos, D. and Rigby, R. (2007). Generalized additive models for location, scale and shape (GAMLSS)., Journal of Statistical Software, 23 1–46. doi:10.18637/jss.v023.i07.
  • [42] Taylor, J. and Verbyla, A. (2004). Joint modeling of location and scale parameters of the t distribution., Statistical Modelling, 4 91–112. doi:10.1191/1471082X04st068oa.
  • [43] Van Keilegom, I. and Wang, L. (2010). Semiparametric modeling and estimation of heteroscedasticity in regression analysis of cross-sectional data., Electronic Journal of Statistics, 4 133–160. doi:10.1214/09-EJS547.
  • [44] Vanegas, L., Rondon, L., and Cysneiros, F. (2012). Diagnostic procedures in Birnbaum-Saunders nonlinear regression models., Computational Statistics and Data Analysis, 56 1662–1680. doi:10.1016/j.csda.2011.10.008.
  • [45] Venezuela, M. K. and Artes, R. (2014). Estimating equations and diagnostic techniques applied to zero-inflated models for panel data., Electronic Journal of Statistics, 8 1641–1660. doi:10.1214/14-EJS936.
  • [46] Villegas, C., Paula, G., and Leiva, V. (2011). Birnbaum-Saunders mixed models for censored reliability data analysis., IEEE Transactions on Reliability, 60 748–758. doi:10.1109/TR.2011.2170251.
  • [47] Wanke, P. and Leiva, V. (2015). Exploring the potential use of the Birnbaum-Saunders distribution in inventory management., Mathematical Problems in Engineering, Article ID 827246:1–9. doi:10.1155/2015/827246.
  • [48] Weisberg, S. (2005)., Applied Linear Regression. Wiley, New York, US. doi:10.1002/0471704091.
  • [49] Wu, L., Zhang, Z., and Xu, D. (2012). Variable selection for joint mean and dispersion models of the lognormal distribution., Hacettepe Journal of Mathematics and Statistics, 41 307–320. http://www.hjms.hacettepe.edu.tr/uploads/b6d36a93-bbda-4e78-a4a7-940cc4d2e31f.pdf.
  • [50] Xie, F. and Wei, B. (2007). Diagnostics analysis for log-Birnbaum-Saunders regression models., Computational Statistics and Data Analysis, 51 4692–4706. doi:10.1016/j.csda.2006.08.030.