Exact tests and confidence regions in nonlinear regression

Abstract

Nonlinear regression models with spherically symmetric error vectors and a single nonlinear parameter are considered. On the basis of a new geometric approach, exact one- and two-sided tests and confidence regions for the nonlinear parameter are derived in the cases of known and unknown error variances. A geometric measure representation formula is used to determine the power functions of the tests if the error variance is known and to derive different lower bounds for the power function of a one-sided test in the case of an unknown error variance. The latter can be done quite effectively by constructing and measuring several balls inside the critical region. A numerical study compares the results for different density generating functions of the error distribution.

Keywords:

• Nonlinear regression
• Spherically symmetric errors
• Exact tests
• Confidence regions
• Power functions
• Exact distributions
• Student distribution
• Doubly non-central generalized t-distribution
• Geometric representation formulae
• Intersection percentage function
• Heavy or light error tails

Acknowledgement

The authors are grateful to A. Pazman and the referees for making several comments to a former version of this article, which lead to improvements and a better presentation of the material.