Robust nonlinear regression in applications

Robust statistical methods, such as M-estimators, are needed for nonlinear regression models because of the presence of outliers/influential observations and heteroscedasticity. Outliers and influential observations are commonly observed in many applications, especially in toxicology and agricultural experiments. For example, dose response studies, which are routinely conducted in toxicology and agriculture, sometimes result in potential outliers, especially in the high dose groups. This is because response to high doses often varies among experimental units (e.g., animals). Consequently, this may result in outliers (i.e., very low values) in that group. Unlike the linear models, in nonlinear models the outliers not only impact the point estimates of the model parameters but can also severely impact the estimate of the information matrix. Note that, the information matrix in a nonlinear model is a function of the model parameters. This is not the case in linear models. In addition to outliers, heteroscedasticity is a major concern when dealing with nonlinear models. Ignoring heteroscedasticity may lead to inaccurate coverage probabilities and Type I error rates. Robustness to outliers/influential observations and to heteroscedasticity is even more important when dealing with thousands of nonlinear regression models in quantitative high throughput screening assays. Recently, these issues have been studied very extensively in the literature (references are provided in this paper), where the proposed estimator is robust to outliers/influential observations as well as to heteroscedasticity. The focus of this paper is to provide the theoretical underpinnings of robust procedures developed recently.