Birnbaum-Saunders models have largely been applied in material fatigue studies and reliability analyses to relate the total time until failure with some type of cumulative damage. In many problems related to the medical field, such as chronic cardiac diseases and different types of cancer, a cumulative damage caused by several risk factors might cause some degradation that leads to a fatigue process. In these cases, BS models can be suitable for describing the propagation lifetime. However, since the cumulative damage is assumed to be normally distributed in the BS distribution, the parameter estimates from this model can be sensitive to outlying observations. In order to attenuate this influence, we present in this paper BS models, in which a Student-t distribution is assumed to explain the cumulative damage. In particular, we show that the maximum likelihood estimates of the Student-t log-BS models attribute smaller weights to outlying observations, which produce robust parameter estimates. Also, some inferential results are presented. In addition, based on local influence and deviance component and martingale-type residuals, a diagnostics analysis is derived. Finally, a motivating example from the medical field is analyzed using log-BS regression models. Since the parameter estimates appear to be very sensitive to outlying and influential observations, the Student-t log-BS regression model should attenuate such influences. The model checking methodologies developed in this paper are used to compare the fitted models.
Generalized Birnbaum-Saunders distribution Likelihood methods Local influence Log-linear models Residual analysis Robustness Sinh-normal distribution
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Balakrishnan N, Leiva V, López J (2007) Acceptance sampling plans from truncated life tests from generalized Birnbaum-Saunders distribution. Commun Stat Simul Comput 36: 643–656CrossRefMATHGoogle Scholar
Berkane M, Kano Y, Bentler PM (1994) Pseudo maximum likelihood estimation in elliptical theory: effects of misspecification. Comput Stat Data Anal 18: 255–267CrossRefMathSciNetGoogle Scholar
Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2. Wiley, New YorkMATHGoogle Scholar
Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data. Wiley, New YorkMATHGoogle Scholar
Klein JP, Moeschberger ML (1997) Survival analysis: techniques for censored and truncated data. Springer, New YorkMATHGoogle Scholar
Lange KL, Little JA, Taylor MG (1989) Robust statistical modelling using the t distribution. J Amer Stat Soc 84: 881–896MathSciNetGoogle Scholar
Lawless JF (2002) Statistical models and methods for lifetime data, 2nd edn. Wiley, New YorkGoogle Scholar
Lee ET, Wang JW (2003) Statistical methods for survival data analysis. Wiley, New YorkMATHGoogle Scholar
Leiva V, Barros M, Paula GA, Galea M (2007) Influence diagnostics in log-Birnbaum-Saunders regression models with censored data. Comput Stat Data Anal 51: 5694–5707CrossRefGoogle Scholar
Leiva V, Barros M, Paula GA, Sanhueza A (2008a) Generalized Birnbaum-Saunders distribution applied to air pollutant concentration. Environmetrics 19 (in press) (doi:10.1002/env.861)
Leiva V, Riquelme M, Balakrishnan N, Sanhueza A (2008b) Lifetime analysis based on the generalized Birnbaum-Saunders distribution. Comput Stat Data Anal 52: 2079–2097CrossRefGoogle Scholar
Leiva V, Sanhueza A, Angulo JM (2008c) A length-biased version of the Birnbaum-Saunders distribution with application in water quality. Stoch Environ Res Risk Assess (in press) (doi:10.1007/s00477-008-0215-9)