Sensitivity analysis for Box-Cox power transformation model: Contrast parameters

Abstract

SUMMARY Sensitivity analysis is an important tool for the evaluation of scientific models. We consider the Box-Cox power transformation model and study the sensitivity for the estimated slope vector when the transformation parameter is perturbed. This is a key issue in the debate among Bickel & Doksum (1981), Box & Cox (1982) and Hinkley & Runger (1984) on the application of the power transformation model. Hinkley & Runger (1984) conjectured that Box & Cox's (1964) z transformation would climinate asymptotically the sensitivity for the estimated slope vector. We establish this conjecture under appropriate symmetry conditions on the joint distribution for the regressors, x . When the true transformation is logarithmic, the conjecture holds if the distribution for x is axially symmetric. For other power transformations, the conjecture holds if x is multivariate normal. We also consider a weaker version of Hinkley & Runger's conjecture, namely, the z transformation would achieve the best possible reduction in the sensitivity. We establish this weaker conjecture under less restrictive symmetry conditions which only involve the marginal distribution for x β. Both conjectures might fail when those symmetry conditions are not satisfied. Two examples are given to illustrate the limitations of the conjectures.