On a new type of Birnbaum-Saunders models and its inference and application to fatigue data. (English) Zbl 1521.62247
Summary: The Birnbaum-Saunders distribution is a widely studied model with diverse applications. Its origins are in the modeling of lifetimes associated with material fatigue. By using a motivating example, we show that, even when lifetime data related to fatigue are modeled, the Birnbaum-Saunders distribution can be unsuitable to fit these data in the distribution tails. Based on the nice properties of the Birnbaum-Saunders model, in this work, we use a modified skew-normal distribution to construct such a model. This allows us to obtain flexibility in skewness and kurtosis, which is controlled by a shape parameter. We provide a mathematical characterization of this new type of Birnbaum-Saunders distribution and then its statistical characterization is derived by using the maximum-likelihood method, including the associated information matrices. In order to improve the inferential performance, we correct the bias of the corresponding estimators, which is supported by a simulation study. To conclude our investigation, we retake the motivating example based on fatigue life data to show the good agreement between the new type of Birnbaum-Saunders distribution proposed in this work and the data, reporting its potential applications.
MSC:
| 62-XX | Statistics |
Keywords:
correction of bias; fatigue life data; maximum likelihood method; Monte Carlo simulation; R software; skew-normal distributionReferences:
| [1] | Arellano-Valle, R. B.; Gómez, H. W.; Quintana, F. A., A new class of skew-normal distributions, Comm. Statist. Theory Methods, 33, 1465-1480 (2004) · Zbl 1134.60304 · doi:10.1081/STA-120037254 |
| [2] | Arellano-Valle, R. B.; Gómez, H. W.; Salinas, H. S., A note on the Fisher information matrix for the skew-generalized-normal model, SORT, 37, 19-28 (2013) · Zbl 1296.62039 |
| [3] | Arrué, J.; Arellano-Valle, R. B.; Gómez, H. W., Bias reduction of maximum likelihood estimates for a modified skew normal distribution, J. Stat. Comput. Simul., 86, 2967-2984 (2016) · Zbl 07184777 · doi:10.1080/00949655.2016.1143471 |
| [4] | Athayde, E.; Azevedo, A.; Barros, M.; Leiva, V., Failure rate of Birnbaum-Saunders distributions: Shape, change-point, estimation and robustness, Braz. J. Probab. Stat., 33, 301-328 (2019) · Zbl 1419.62025 · doi:10.1214/17-BJPS389 |
| [5] | Athayde, E.; Azevedo, A.; Leiva, V.; Sanhueza, A., About Birnbaum-Saunders distributions based on the Johnson system, Comm. Statist. Theory Methods, 41, 2061-2079 (2012) · Zbl 1251.60013 · doi:10.1080/03610926.2010.551454 |
| [6] | Azzalini, A., A class of distributions which includes the normal ones, Scand. J. Statist., 11, 171-178 (1985) · Zbl 0581.62014 |
| [7] | Balakrishnan, N.; Gupta, R.; Kundu, D.; Leiva, V.; Sanhueza, A., On some mixture models based on the Birnbaum-Saunders distribution and associated inference, J. Stat. Plan. Inference, 141, 2175-2190 (2011) · Zbl 1214.62014 · doi:10.1016/j.jspi.2010.12.005 |
| [8] | Balamurali, S.; Jeyadurga, P.; Usha, M., Optimal design of repetitive group sampling plans for Weibull and gamma distributions with applications and comparison to the Birnbaum-Saunders distribution, J. Appl. Stat., 45, 2499-2520 (2018) · Zbl 1516.62141 · doi:10.1080/02664763.2018.1426740 |
| [9] | Bhatti, C., The Birnbaum-Saunders autoregressive conditional duration model, Math. Comput. Simul., 80, 2062-2078 (2010) · Zbl 1195.62136 · doi:10.1016/j.matcom.2010.01.011 |
| [10] | Birnbaum, Z. W.; Saunders, S. C., A new family of life distributions, J. Appl. Probab., 6, 319-327 (1969) · Zbl 0209.49801 · doi:10.2307/3212003 |
| [11] | Birnbaum, Z. W.; Saunders, S. C., Estimation for a family of life distributions with applications to fatigue, J. Appl. Probab., 6, 328-347 (1969) · Zbl 0216.22702 · doi:10.2307/3212004 |
| [12] | Castillo, N. O.; Gómez, H. W.; Bolfarine, H., Epsilon Birnbaum-Saunders distribution family: Properties and inference, Statist. Papers, 52, 871-883 (2011) · Zbl 1284.60026 · doi:10.1007/s00362-009-0293-x |
| [13] | Cox, D. R.; Reid, N., Parameter orthogonality and approximate conditional inference, J. R. Stat. Soc. B, 49, 1-39 (1987) · Zbl 0616.62006 |
| [14] | Cox, D. R.; Snell, E., A general definition of residuals, J. R. Stat. Soc. B, 2, 248-275 (1968) · Zbl 0164.48903 |
| [15] | Desousa, M. F.; Saulo, H.; Leiva, V.; Scalco, P., On a tobit-Birnbaum-Saunders model with an application to antibody response to vaccine, J. Appl. Stat., 45, 932-955 (2018) · Zbl 1516.62245 · doi:10.1080/02664763.2017.1322559 |
| [16] | Ferreira, M.; Gomes, M. I.; Leiva, V., On an extreme value version of the Birnbaum-Saunders distribution, Revstat Stat. J., 10, 181-210 (2012) · Zbl 1297.60006 |
| [17] | Firth, D., Bias reduction of maximum likelihood estimates, Biometrika, 80, 27-38 (1993) · Zbl 0769.62021 · doi:10.1093/biomet/80.1.27 |
| [18] | Fonseca, R. V.; Cribari-Neto, F., Bimodal Birnbaum-Saunders generalized autoregressive score model, J. Appl. Stat., 45, 2585-2606 (2018) · Zbl 1516.62287 · doi:10.1080/02664763.2018.1428734 |
| [19] | Garcia-Papani, F.; Leiva, V.; Ruggeri, F.; Uribe-Opazo, M. A., Kriging with external drift in a Birnbaum-Saunders geostatistical model, Stoch. Environ. Res. Risk Assess., 32, 1517-1530 (2018) · doi:10.1007/s00477-018-1546-9 |
| [20] | Garcia-Papani, F.; Leiva, V.; Uribe-Opazo, M. A.; Aykroyd, R. G., Birnbaum-Saunders spatial regression models: Diagnostics and application to chemical data, Chemometr. Intell. Lab. Syst., 177, 114-128 (2018) · doi:10.1016/j.chemolab.2018.03.012 |
| [21] | Garcia-Papani, F.; Uribe-Opazo, M. A.; Leiva, V.; Aykroyd, R. G., Birnbaum-Saunders spatial modelling and diagnostics applied to agricultural engineering data, Stoch. Environ. Res. Risk Assess., 31, 105-124 (2017) · doi:10.1007/s00477-015-1204-4 |
| [22] | Gómez, H. W.; Olivares-Pacheco, J. F.; Bolfarine, H., An extension of the generalized Birnbaum-Saunders distribution, Statist. Probab. Lett., 79, 331-338 (2009) · Zbl 1155.62006 · doi:10.1016/j.spl.2008.08.014 |
| [23] | Guo, X.; Wu, H.; Li, G.; Li, Q., Inference for the common mean of several Birnbaum-Saunders populations, J. Appl. Stat., 44, 941-954 (2017) · Zbl 1516.62313 · doi:10.1080/02664763.2016.1189521 |
| [24] | Huerta, M.; Leiva, V.; Liu, S.; Rodriguez, M.; Villegas, D., On a partial least squares regression model for asymmetric data with a chemical application in minings, Chemometr. Intell. Lab. Syst., 190, 55-68 (2019) · doi:10.1016/j.chemolab.2019.04.013 |
| [25] | Jin, X.; Kawczak, J., Birnbaum-Saunders and lognormal kernel estimators for modelling durations in high frequency financial data, Ann. Econ. Finance, 4, 103-124 (2003) |
| [26] | Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous Univariate Distributions, 2 (1995), Wiley: Wiley, New York, US · Zbl 0821.62001 |
| [27] | Kotz, S.; Leiva, V.; Sanhueza, A., Two new mixture models related to the inverse Gaussian distribution, Methodol. Comput. Appl. Probab., 12, 199-212 (2010) · Zbl 1187.60009 · doi:10.1007/s11009-008-9112-4 |
| [28] | Kundu, D.; Kannan, N.; Balakrishnan, N., On the hazard function of Birnbaum-Saunders distribution and associated inference, Comput. Stat. Data Anal., 52, 2692-2702 (2008) · Zbl 1452.62729 · doi:10.1016/j.csda.2007.09.021 |
| [29] | Leão, J.; Leiva, V.; Saulo, H.; Tomazella, V., Birnbaum-Saunders frailty regression models: Diagnostics and application to medical data, Biom. J., 59, 291-314 (2017) · Zbl 1367.62293 · doi:10.1002/bimj.201600008 |
| [30] | Leão, J.; Leiva, V.; Saulo, H.; Tomazella, V., A survival model with Birnbaum-Saunders frailty for uncensored and censored cancer data, Braz. J. Probab. Statist., 32, 707-729 (2018) · Zbl 1404.62125 · doi:10.1214/17-BJPS360 |
| [31] | Leão, J.; Leiva, V.; Saulo, H.; Tomazella, V., Incorporation of frailties into a cure rate regression model and its diagnostics and application to melanoma data, Stat. Med., 37, 4421-4440 (2018) · doi:10.1002/sim.7929 |
| [32] | Leiva, V., The Birnbaum-Saunders Distribution (2016), Academic Press: Academic Press, New York, US · Zbl 1338.62010 |
| [33] | Leiva, V.; Athayde, E.; Azevedo, C.; Marchant, C., Modeling wind energy flux by a Birnbaum-Saunders distribution with unknown shift parameter, J. Appl. Stat., 38, 2819-2838 (2011) · Zbl 1511.62272 · doi:10.1080/02664763.2011.570319 |
| [34] | Leiva, V.; Marchant, C.; Saulo, H.; Aslam, M.; Rojas, F., Capability indices for Birnbaum-Saunders processes applied to electronic and food industries, J. Appl. Stat., 41, 1881-1902 (2014) · Zbl 1352.62170 · doi:10.1080/02664763.2014.897690 |
| [35] | Lillo, C.; Leiva, V.; Nicolis, O.; Aykroyd, R. G., L-moments of the Birnbaum-Saunders distribution and its extreme value version: Estimation, goodness of fit and application to earthquake data, J. Appl. Stat., 45, 187-209 (2018) · Zbl 1516.62415 · doi:10.1080/02664763.2016.1269729 |
| [36] | Liseo, B., The skew-normal class of densities: Inferential aspects from a Bayesian viewpoint, Statistica, 50, 59-70 (1990) |
| [37] | Marchant, C.; Leiva, V.; Cysneiros, F. J.A., A multivariate log-linear model for Birnbaum-Saunders distributions, IEEE Trans. Reliab., 65, 816-827 (2016) · doi:10.1109/TR.2015.2499964 |
| [38] | Marchant, C.; Leiva, V.; Christakos, G.; Cavieres, M. A., Monitoring urban environmental pollution by bivariate control charts: New methodology and case study in Santiago, Chile, Environmetrics, 30, e2551 (2019) · doi:10.1002/env.2551 |
| [39] | Marchant, C.; Leiva, V.; Cysneiros, F. J.A.; Liu, S., Robust multivariate control charts based on Birnbaum-Saunders distributions, J. Stat. Comput. Simul., 88, 182-202 (2018) · Zbl 07192547 · doi:10.1080/00949655.2017.1381699 |
| [40] | Marchant, C.; Leiva, V.; Cysneiros, F. J.A.; Vivanco, J. F., Diagnostics in multivariate generalized Birnbaum-Saunders regression models, J. Appl. Stat., 43, 2829-2849 (2016) · Zbl 1516.62459 · doi:10.1080/02664763.2016.1148671 |
| [41] | Martínez-Flórez, G.; Bolfarine, H.; Gómez, H. W., An alpha-power extension for the Birnbaum-Saunders distribution, Statistics, 48, 896-912 (2014) · Zbl 1326.62029 · doi:10.1080/02331888.2013.846910 |
| [42] | Olmos, N. M.; Martínez-Flórez, G.; Bolfarine, H., Bimodal Birnbaum-Saunders distribution with applications to non negative measurements, Comm. Statist. Theory Methods, 46, 6240-6257 (2017) · Zbl 1368.60016 · doi:10.1080/03610926.2015.1133824 |
| [43] | Paula, G. A.; Leiva, V.; Barros, M.; Liu, S., Robust statistical modeling using the Birnbaum-Saunders-t distribution applied to insurance, Appl. Stoch. Models Bus. Ind., 28, 16-34 (2012) · Zbl 06292429 · doi:10.1002/asmb.887 |
| [44] | R Core Team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2017. |
| [45] | Rousseeuw, P.J., Croux, C., Todorov, V., Ruckstuhl, A., Salibian-Barrera, M., Verbeke, T., Koller, M., and Maechler, M., robustbase: Basic robust statistics. R package version 0.92-6 (2016). |
| [46] | Santos-Neto, M.; Cysneiros, F. J.A.; Leiva, V.; Barros, M., Reparameterized Birnbaum-Saunders regression models with varying precision, Electron. J. Stat., 10, 2825-2855 (2016) · Zbl 1348.62220 · doi:10.1214/16-EJS1187 |
| [47] | Sartori, N., Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions, J. Stat. Plan. Inference, 136, 4259-4275 (2006) · Zbl 1098.62023 · doi:10.1016/j.jspi.2005.08.043 |
| [48] | Saulo, H.; Leao, J.; Leiva, V.; Aykroyd, R. G., Birnbaum-Saunders autoregressive conditional duration models applied to high-frequency financial data, Statist. Papers (2019) · Zbl 1432.62315 |
| [49] | Saulo, H.; Leiva, V.; Ziegelmann, F. A.; Marchant, C., A nonparametric method for estimating asymmetric densities based on skewed Birnbaum-Saunders distributions applied to environmental data, Stoch. Environ. Res. Risk Assess., 27, 1479-1491 (2013) · doi:10.1007/s00477-012-0684-8 |
| [50] | Villegas, C.; Paula, G. A.; Leiva, V., Birnbaum-Saunders mixed models for censored reliability data analysis, IEEE Trans. Reliab., 60, 748-758 (2011) · doi:10.1109/TR.2011.2170251 |
| [51] | Wanke, P.; Leiva, V., Exploring the potential use of the Birnbaum-Saunders distribution in inventory management, Math. Probl. Eng., 2015, 1-9 (2015) · Zbl 1394.90057 · doi:10.1155/2015/827246 |
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