We consider the Box-Cox power transformation model and study the sensitivity for the estimated slope vector when the transformation parameter is perturbed. D. V. Hinkley and G. Runger [J. Am. Stat. Assoc. 79, 302-320 (1984; Zbl 0553.62051)] conjectured that G. E. P. Box and D. R. Cox’s [J. R. Stat. Soc., Ser. B 26, 211-243 (1964; Zbl 0156.401)] \(z\) transformation would eliminate 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 and 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 \(x\beta\). Both conjectures might fail when those symmetry conditions are not satisfied. Two examples are given to illustrate the limitation of the conjectures.