IJPAM: Volume 21, No. 4 (2005)

ROBUST ESTIMATION AND TESTING IN
NONLINEAR REGRESSION MODELS

Ruei-Che Liu

, Marianthi Markatou

, Chih-Ling Tsai

PsychoGenics, Inc., Hawthorne, NY 10532
e-mail: rl241@columbia.edu

Department of Biostatistics
722 West 168th Street, Floor 6
Mailman School of Public Health
Columbia University, New York, NY 10032
e-mail: mm168@columbia.edu

Graduate School of Management
One Shields Avenue
University of California
Davis, CA 95616
e-mail: cltsai@ucdavis.edu

Abstract.We study the performance of least squares estimators, robust estimators, and tests in nonlinear regression models under various contamination schemes of the error distribution. We also address the problem of obtaining appropriate initial values for the algorithms that compute the M-estimators. It is shown that a scheme proposed by Smyth in the context of least squares offers good initial values for the computation of M-estimators. The performance of the estimators is measured in terms of their bias and variance, while that of the tests is measured in terms of attaining the nominal level. It is shown that for symmetric contamination with relatively low variance, the nonlinear least squares estimators perform as well as the M-estimators in terms of bias, however their standard deviation is larger than that of the M-estimators. The M-estimators perform well under small levels of both symmetric and asymmetric contamination. In general, the performance of the estimators depends on the nonlinear regression function that is fitted, and the effect of certain types of contamination is more pronounced on the variance of the estimates rather than on their bias.

The tests based on M-estimators have a level close to the nominal level, even for relatively high percentages of symmetric contamination with relatively low variance. Moreover, for the cases studied, with the subhypothesis testing problem $\theta_{2}=0$ , $\theta_{1}$ unspecified, $\theta=(\theta_{1}% ^{T},\theta_{2}^{T})^{T}$ , it is shown that the chi-squared with $p_{2}% =$ dim( $\theta_{2}$ ) degrees of freedom fits the quantiles of the Wald test, while the $F(p_{2},N-p)$ distribution fits the drop-in-dispersion test, where dim( $\theta$ ), N is the sample size.