Transformations for Improving Linearization Confidence Intervals in Nonlinear Regression

原文摘要

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 have poor coverage probabilities. Using gradient direction plots, we identify transformations of g(θ) that lead to more accurate LA intervals. These include power transformations, whose effectiveness is demonstrated in a variety of nonlinear regression problems via a simulation study. Finally, we show how to use profile t plots and bias indices to suggest transforms to improve LA intervals. The idea is to find a monotone transformation T such that the linearization confidence interval for T(g(θ)) has coverage probability close to its nominal value and then invert this interval to give an accurate interval for g(θ). The transformation T is obtained from a gradient direction plot that may be thought of as an attempt to view the graph of the estimator ĝ of g(θ) in two dimensions. We develop theory to motivate this procedure and identify conditions under which the intervals produced are exact. We use the simulation study to demonstrate that the intervals also work well when the conditions of the theory are not satisfied, as is the usual case in practice. By showing how the profile t plot and bias indices are related to the gradient direction plot, we show how these may also be used to suggest approximate transformations T. Because, as is shown, linear approximation intervals are invariant under reparameterization, our development is based on invariant constructs such as g, the solution locus M for the regression model, and the gradient direction plot.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）