ON TESTING THE GOODNESS-OF-FIT OF NONLINEAR HETEROSCEDASTIC REGRESSION MODELS

Abstract

For testing the adequacy of a parametric model in regression, various test statistics can be constructed on the basis of a marked empirical process of residuals. By using a discretized version of the decomposition of the corresponding Gaussian limiting process into its principal components, we obtain a test statistic with an asymptotic chi-squared distribution under the null hypothesis. We investigate the consistency of this test statistic and of the estimators needed to compute it. Numerical experiments indicate that the distributional approximations already work for small to moderate sample sizes and reveal that the test has good power properties against a variety of alternatives. The test has a simple implementation. We present an application to a real-data example for testing the adequacy of a possible heteroscedastic exponential model.

Keywords:

• Exponential regression model
• Gaussian process
• Goodness-of-fit
• Nonlinear regression
• Principal components

ACKNOWLEDGMENTS

The second author, who was supported by a grant from the Swiss National Science Foundation, acknowledges the hospitality of La Trobe University where this paper was written. The authors thank Professors R. G. Staudte and W. Stute for their helpful comments and suggestions.