Diagnostics for Linearization Confidence Intervals in Nonlinear Regression

Abstract

We investigate linear approximation (LA) confidence intervals for functions g(θ) of the parameters θ in a nonlinear regression model. These intervals are almost universally used and generally perform well, but at times they have poor coverage probabilities. A diagnostic plot and index are developed to detect these failures. We show how these diagnostics may be used to estimate coverage probabilities and these are used to calibrate the diagnostics. The performance of the coverage probability estimates in a variety of nonlinear regression problems is investigated via simulation; for these problems, they work quite well. Conditions are identified under which the estimates are exact. Finally, we discuss the use of the profile t plot and asymmetry and bias indices as diagnostics for LA intervals and show how to calibrate them in terms of coverage probabilities.

Key Words:

• Coverage probability approximation
• Estimator differential
• Generalized linear function
• Gradient direction plot
• Invariance
• Nonlinearity index