Designing to detect heteroscedasticity in a regression model

TL;DR

This paper develops experimental design strategies to detect heteroscedasticity in non-linear Gaussian regression models, linking design criteria with the power of likelihood-based tests, and confirming findings through simulations.

Contribution

It establishes the asymptotic relationship between ${ m D}_s$- and KL-criteria and the noncentrality parameter for heteroscedasticity detection, extending to multi-parameter variance functions.

Findings

When variance depends on one parameter, criteria coincide asymptotically.

KL-optimum design maximizes the noncentrality parameter for multi-parameter variance functions.

Simulation confirms theoretical relationships and power calculations.

Abstract

We consider the problem of designing experiments to detect the presence of a specified heteroscedastity in a non-linear Gaussian regression model. In this framework, we focus on the ${\rm D}_s$- and KL-criteria and study their relationship with the noncentrality parameter of the asymptotic chi-squared distribution of a likelihood-based test, for local alternatives. Specifically, we found that when the variance function depends just on one parameter, the two criteria coincide asymptotically and in particular, the ${\rm D}_1$-criterion is proportional to the noncentrality parameter. Differently, if the variance function depends on a vector of parameters, then the KL-optimum design converges to the design that maximizes the noncentrality parameter. Furthermore, we confirm our theoretical findings through a simulation study concerning the computation of asymptotic and exact powers of the log-likelihood ratio statistic.