The application of the Box-Tidwell transformation in reliability modeling

The Box-Tidwell represents a commonly-used iterative approach in linear or nonlinear regression but is little used in reliabiity modeling. It provides a power transformation of the regressor variable in order to linearize the model, or (occasionally) defaults to a log transformation. Its main drawback is lack of convergence under certain circumstances which results in it recommending a log transformation inappropriately. The techniques developed in this paper significantly increase the Box-Tidwell's robustness and ensure a power transformation solution is consistently found. Used along with weighted least squares (WLS), the Box-Tidwell transformation represents a real alternative to maximum likelihood or graphical estimation. This paper takes the example of the power-law model used in reliability growth analysis and demonstrates the application and effectiveness of the robust Box-Tidwell. Extensive simulation modelling has shown it generally provides a better fit to the data than the alternative maximum likelihood estimates. It illustrates that maximum likelihood methods do not always provide the 'best' estimator in the sense of one that minimzes a suitable loss function. The comparative analysis was conducted using simulation of 10, 30, and 100 observations for the power-law model. Cross validation was conducted using the Predicted Residual Sum of Squares (PRESS) statistic. Contrary to expectation, the PRESS statistics shows that the parameter estimation by this methodology (called BTW) will provide the best fit to the data (in the sense of minimizing the sum of the squared errors), and not estimation by maximum likelihood methods. The BTW will provide the best interpolated predictions compared to the alternatives.