Robust estimation of the Birnbaum-Saunders distribution

The Birnbaum-Saunders distribution is prevalent in the engineering sciences as an effective means of modeling fatigue life. In practice however, there is no guarantee that the collected data follow such a model. Consequently, this paper considers the robust estimation of the parameters & quantiles of this distribution. This robust procedure is a powerful alternative to commonly used procedures, such as MLE (maximum-likelihood estimate), which are sensitive to model deviations that often occur in practice hence yielding severely distorted parameter estimates. Our robust estimation technique is based on OBRE (optimal bias-robust estimator) and assigns a weight to each observation and gives estimates of the parameters and quantiles based on data which are well modeled by the distribution. Thus, observations which are not consistent with the proposed distribution can be identified and the validity of the model assessed. An 'application to aluminum fatigue data' and 'simulation results' provide strong evidence in support of OBRE. OBRE is extremely competitive with the MLE at the model. As well, in the presence of bad data, OBRE provides good estimates with modest standard deviations based on 'the bulk of the data' and 'insightful weights identifying observations lying outside the model'. The OBRE efficiency deteriorates with increasing a (shape parameter) but for the usual range of 0 < alpha < 1, OBRE performs more than adequately for practical purposes. Furthermore, efficiency in many ways becomes a non-issue as we move away from the model. We must give up some degree of efficiency to gain robustness, and OBRE provides a powerful method of doing so. The simulation study shows that compromises can be made which are effective in both regards. Since statistical-confidence intervals can be calculated for OBRE, robust statistical-confidence interval estimates for the critical time of the hazard rate can also be obtained. These techniques are fundamental in describing, analyzing, and comparing fatigue data so that engineers can achieve the desired reliability on a rational basis and at the same time avoid serious consequences stemming from incorrect inference.