Brazilian Journal of Probability and Statistics

Failure rate of Birnbaum–Saunders distributions: Shape, change-point, estimation and robustness

Emilia Athayde, Assis Azevedo, Michelli Barros, and Víctor Leiva

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The Birnbaum–Saunders (BS) distribution has been largely studied and applied. A random variable with BS distribution is a transformation of another random variable with standard normal distribution. Generalized BS distributions are obtained when the normally distributed random variable is replaced by another symmetrically distributed random variable. This allows us to obtain a wide class of positively skewed models with lighter and heavier tails than the BS model. Its failure rate admits several shapes, including the unimodal case, with its change-point being able to be used for different purposes. For example, to establish the reduction in a dose, and then in the cost of the medical treatment. We analyze the failure rates of generalized BS distributions obtained by the logistic, normal and Student-t distributions, considering their shape and change-point, estimating them, evaluating their robustness, assessing their performance by simulations, and applying the results to real data from different areas.

Article information

Source
Braz. J. Probab. Stat., Volume 33, Number 2 (2019), 301-328.

Dates
Received: February 2016
Accepted: November 2017
First available in Project Euclid: 4 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1551690036

Digital Object Identifier
doi:10.1214/17-BJPS389

Keywords
Bootstrapping likelihood-based methods logistic normal and Student-t distributions Monte Carlo simulation R computer language

Citation

Athayde, Emilia; Azevedo, Assis; Barros, Michelli; Leiva, Víctor. Failure rate of Birnbaum–Saunders distributions: Shape, change-point, estimation and robustness. Braz. J. Probab. Stat. 33 (2019), no. 2, 301--328. doi:10.1214/17-BJPS389. https://projecteuclid.org/euclid.bjps/1551690036


Export citation

References

  • Aarset, M. (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability 36, 106–108.
  • Athayde, E. (2017). A characterization of the generalized Birnbaum–Saunders distribution. REVSTAT Statistical Journal 15, 333–354.
  • Azevedo, C., Leiva, V., Athayde, E. and Balakrishnan, N. (2012). Shape and change point analyses of the Birnbaum–Saunders-t hazard rate and associated estimation. Computational Statistics & Data Analysis 56, 3887–3897.
  • Balakrishnan, N. (2013). Handbook of the Logistic Distribution. Boca Raton, US: CRC Press.
  • Barros, M., Leiva, V., Ospina, R. and Tsuyuguchi, A. (2014). Goodness-of-fit tests for the Birnbaum–Saunders distribution with censored reliability data. IEEE Transactions on Reliability 63, 543–554.
  • Barros, M., Paula, G. A. and Leiva, V. (2009). An R implementation for generalized Birnbaum–Saunders distributions. Computational Statistics & Data Analysis 53, 1511–1528.
  • Bebbington, M., Lai, C. and Zitikis, R. (2008). A proof of the shape of the Birnbaum–Saunders hazard rate function. The Mathematical Scientist 33, 49–56.
  • Bhatti, C. (2010). The Birnbaum–Saunders autoregressive conditional duration model. Mathematics and Computers in Simulation 80, 2062–2078.
  • Bourguignon, M., Leão, J., Leiva, V. and Santos-Neto, M. (2017). The transmuted Birnbaum–Saunders distribution. REVSTAT Statistical Journal 15, 601–628.
  • Chang, D. and Tang, L. (1993). Reliability bounds and critical time for the Birnbaum–Saunders distribution. IEEE Transactions on Device and Materials Reliability 42, 464–469.
  • Chen, G. and Balakrishnan, N. (1995). A general purpose approximate goodness-of-fit test. Journal of Quality Technology 27, 154–161.
  • D’Agostino, C. and Stephens, M. (1986). Goodness-of-Fit Techniques. New York, US: Marcel Dekker.
  • Desousa, M. F., Saulo, H., Leiva, V. and Scalco, P. (2018). On a tobit-Birnbaum–Saunders model with an application to antibody response to vaccine. Journal of Applied Statistics 45, 932–955.
  • Dupuis, D. J. and Mills, J. E. (1998). Robust estimation of the Birnbaum–Saunders distribution. IEEE Transactions on Reliability 1, 88–95.
  • Engelhardt, M., Bain, L. and Wright, F. (1981). Inferences on the parameters of the Birnbaum–Saunders fatigue life distribution based on maximum likelihood estimation. Technometrics 23, 251–256.
  • 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.
  • Garcia-Papani, F., Uribe-Opazo, M. A., Leiva, V. and Aykroyd, R. G. (2017). Birnbaum–Saunders spatial modelling and diagnostics applied to agricultural engineering data. Stochastic Environmental Research and Risk Assessment 31, 105–124.
  • Kundu, D. (2015a). Bivariate log-Birnbaum–Saunders distribution. Statistics 49, 900–917.
  • Kundu, D. (2015b). Bivariate sinh-normal distribution and a related model. Brazilian Journal of Probability and Statistics 20, 590–607.
  • Kundu, D., Kannan, N. and Balakrishnan, N. (2008). On the hazard function of Birnbaum–Saunders distribution and associated inference. Computational Statistics & Data Analysis 52, 2692–2702.
  • Leão, J., Leiva, V., Saulo, H. and Tomazella, V. (2017). Birnbaum–Saunders frailty regression models: Diagnostics and application to medical data. Biometrical Journal 59, 291–314.
  • Leão, J., Leiva, V., Saulo, H. and Tomazella, V. (2018a). Incorporation of frailties into a cure rate regression model and its diagnostics and application to melanoma data. Statistics in Medicine 37, 4421–4440.
  • Leão, J., Leiva, V., Saulo, H. and Tomazella, V. (2018b). A survival model with Birnbaum–Saunders frailty for uncensored and censored cancer data. Brazilian Journal of Probability and Statistics 32, 707–729.
  • Leiva, V., Athayde, E., Azevedo, C. and Marchant, C. (2011). Modeling wind energy flux by a Birnbaum–Saunders distribution with unknown shift parameter. Journal of Applied Statistics 38, 2819–2838.
  • Leiva, V., Ferreira, M., Gomes, M. and Lillo, C. (2016a). Extreme value Birnbaum–Saunders regression models applied to environmental data. Stochastic Environmental Research and Risk Assessment 30, 1045–1058.
  • 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.
  • Leiva, V., Marchant, C., Saulo, H., Aslam, M. and Rojas, F. (2014a). Capability indices for Birnbaum–Saunders processes applied to electronic and food industries. Journal of Applied Statistics 41, 1881–1902.
  • Leiva, V., Rojas, E., Galea, M. and Sanhueza, A. (2014b). Diagnostics in Birnbaum–Saunders accelerated life models with application to fatigue data. Applied Stochastic Models in Business and Industry 30, 115–131.
  • Leiva, V., Ruggeri, F., Saulo, H. and Vivanco, J. F. (2017). A methodology based on the Birnbaum–Saunders distribution for reliability analysis applied to nano-materials. Reliability Engineering & Systems Safety 157, 192–201.
  • Leiva, V., Sanhueza, A., Sen, P. K. and Araneda, N. (2010). M-procedures in the general multivariate nonlinear regression model. Pakistan Journal of Statistics 26, 1–13.
  • Leiva, V., Santos-Neto, M., Cysneiros, F. J. and Barros, M. (2014c). Birnbaum–Saunders statistical modelling: A new approach. Statistical Modelling 14, 21–48.
  • Leiva, V., Santos-Neto, M., Cysneiros, F. J. and Barros, M. (2016b). A methodology for stochastic inventory models based on a zero-adjusted Birnbaum–Saunders distribution. Applied Stochastic Models in Business and Industry 32, 74–89.
  • 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.
  • Lemonte, A. (2013). A new extension of the Birnbaum Saunders distribution. Brazilian Journal of Probability and Statistics 27, 133–149.
  • Lillo, C., Leiva, V., Nicolis, O. and Aykroyd, R. G. (2018). L-moments of the Birnbaum–Saunders distribution and its extreme value version: Estimation, goodness of fit and application to earthquake data. Journal of Applied Statistics 45, 187–209.
  • Lucas, A. (1997). Robustness of the student $t$ based M-estimator. Communications in Statistics Theory and Methods 41, 1165–1182.
  • Marchant, C., Bertin, K., Leiva, V. and Saulo, H. (2013). Generalized Birnbaum–Saunders kernel density estimators and an analysis of financial data. Computational Statistics & Data Analysis 63, 1–15.
  • Marchant, C., Leiva, V. and Cysneiros, F. J. (2016a). A multivariate log-linear model for Birnbaum–Saunders distributions. IEEE Transactions on Reliability 65, 816–827.
  • Marchant, C., Leiva, V., Cysneiros, F. J. and Liu, S. (2018). Robust multivariate control charts based on Birnbaum–Saunders distributions. Journal of Statistical Computation and Simulation 88, 182–202.
  • Marchant, C., Leiva, V., Cysneiros, F. J. and Vivanco, J. F. (2016b). Diagnostics in multivariate generalized Birnbaum–Saunders regression models. Journal of Applied Statistics 43, 2829–2849.
  • Marshall, A. and Olkin, I. (2007). Life Distributions. New York, US: Springer.
  • Ng, H. K., Kundu, D. and Balakrishnan, N. (2003). Modified moment estimation for the two-parameter Birnbaum–Saunders distribution. Computational Statistics & Data Analysis 43, 283–298.
  • 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.
  • Sánchez, L., Leiva, V., Caro-Lopera, F. and Cysneiros, F. J. (2015). On matrix-variate Birnbaum–Saunders distributions and their estimation and application. Brazilian Journal of Probability and Statistics 29, 790–812.
  • Sanhueza, A., Leiva, V. and Balakrishnan, N. (2008). The generalized Birnbaum–Saunders distribution and its theory, methodology and application. Communications in Statistics Theory and Methods 37, 645–670.
  • Santana, L., Vilca, F. and Leiva, V. (2011). Influence analysis in skew-Birnbaum–Saunders regression models and applications. Journal of Applied Statistics 38, 1633–1649.
  • Santos-Neto, M., Cysneiros, F. J., Leiva, V. and Barros, M. (2016). Reparameterized Birnbaum–Saunders regression models with varying precision. Electronic Journal of Statistics 10, 2825–2855.
  • Saulo, H., Leão, J., Leiva, V. and Aykroyd, R. G. (2019). Birnbaum–Saunders autoregressive conditional duration models applied to high-frequency financial data. Statistical Papers. To appear. DOI:dx.doi.org/10.1007/s00362-017-0888-6.
  • 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.
  • Vanegas, L. H. and Paula, G. A. (2015). A semiparametric approach for joint modeling of median and skewness. Test 24, 110–135.
  • Vanegas, L. H. and Paula, G. A. (2016). Log-symmetric distributions: Statistical properties and parameter estimation. Brazilian Journal of Probability and Statistics 30, 196–220.
  • Vilca, F., Balakrishnan, N. and Zeller, C. (2014). The bivariate sinh-elliptical distribution with applications to Birnbaum–Saunders distribution and associated regression and measurement error models. Computational Statistics & Data Analysis 80, 1–16.
  • Villegas, C., Paula, G. A. and Leiva, V. (2011). Birnbaum–Saunders mixed models for censored reliability data analysis. IEEE Transactions on Reliability 60, 748–758.
  • Wang, M., Park, C. and Sun, X. (2015). Simple robust parameter estimation for the Birnbaum–Saunders distribution. Journal of Statistical Distributions and Applications 2, 1–11.
  • Wang, M., Zhao, J., Sun, X. and Park, C. (2013). Robust explicit estimation of the two-parameter Birnbaum–Saunders distribution. Journal of Applied Statistics 40, 2259–2274.
  • 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.