Robust Nonlinear Regression

1Robust Statistics and its Application in Linear Regression

This is an introductory chapter giving the mathematical background to the robust statistics that are used in the rest of the book. Robust linear regression methods are then generalized to nonlinear regression in the rest of the book.

The robust approach to linear regression is described in this chapter. It is the main motivation for extending statistical inference approaches used in linear regression to nonlinear regression. This is done by considering the gradient of a nonlinear model as the design matrix in a linear regression. Outlier detection methods used in linear regression are also extended to use in nonlinear regression.

In this chapter the consistency and asymptotic distributions of robust estimators and robust linear regression are presented. The validity of the results requires certain regularity conditions, which are presented here. Proofs of the theorems are very technical and since this book is about nonlinear regression, they have been omitted.

1.1 Robust Aspects of Data

Robust statistics were developed to interpret data for which classical assumptions, such as randomness, independence, distribution models, prior assumptions about parameters and other prior hypotheses do not apply. Robust statistics can be used in a wide range of problems.

The classical approach in statistics assumes that data are collected from a distribution function; that is, the observed values follow the simultaneous distribution function ...