In this paper we introduce new robust estimators for the logistic and probit regressions for binary, multinomial, nominal and ordinal data and apply these models to estimate the parameters when outliers or inluential observations are present. Maximum likelihood estimates don't behave well when outliers or inluential observations are present. One remedy is to remove inluential observations from the data and then apply the maximum likelihood technique on the deleted data. Another approach is to employ a robust technique that can handle outliers and inluential observations without removing any observations from the data sets. The robustness of the method is tested using real and simulated data sets.

Ordinary least squares estimates can behave badly when outliers are present. An alternative is to use a robust regression technique that can handle outliers and influential observations. We introduce a new robust estimation method called TELBS robust regression method. We also introduce a new measurement called S h (i) for detecting influential observations. In addition, a new measure for goodness of fit, called R 2 RFPR , is introduced. We provide an algorithm to perform the TELBS estimation of regression parameters. Real and simulated data sets are used to assess the performance of this new estimator. In simulated data with outliers, the TELBS estimator of regression parameters performs better in comparison with least squares, M and MM estimators, with respect to both bias and mean squared error. For rat liver weights data, none of the estimators (least squares, M, and MM) are able to estimate the parameters accurately. However, TELBS does give an accurate estimate. Using real data for brain imaging, the TELBS and MM methods were equally accurate. In both of these real data sets, the S h (i) measure was very effective in identifying influential observations. The robustness and simplicity of computations of TELBS model parameters make this method an appropriate one for analysis of linear regression. Algorithms and programs have been provided for ease in implementation, including all relevant statistics necessary to perform a complete analysis of linear regression.

Background When outliers are present, the least squares method of nonlinear regression performs poorly. The main purpose of this paper is to provide a robust alternative technique to the Ordinary Least Squares nonlinear regression method. This new robust nonlinear regression method can provide accurate parameter estimates when outliers and/or influential observations are present. Method Real and simulated data for drug concentration and tumor size-metastasis are used to assess the performance of this new estimator. Monte Carlo simulations are performed to evaluate the robustness of our new method in comparison with the Ordinary Least Squares method. Results In simulated data with outliers, this new estimator of regression parameters seems to outperform the Ordinary Least Squares with respect to bias, mean squared errors, and mean estimated parameters. Two algorithms have been proposed. Additionally and for the sake of computational ease and illustration, a Mathematica program has been provided in the Appendix. Conclusion The accuracy of our robust technique is superior to that of the Ordinary Least Squares. The robustness and simplicity of computations make this new technique more appropriate and useful tool for the analysis of nonlinear regressions.