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Biometrika 1993 80(4):885-897; doi:10.1093/biomet/80.4.885
© 1993 by Biometrika Trust
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Sensitivity analysis for Box-Cox power transformation model: Contrast parameters

NAIHUA DUAN

RAND Corporation Santa Monica, California 90407, U. S.A.

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 xß. Both conjectures might fail when those symmetry conditions are not satisfied. Two examples are given to illustrate the limitations of the conjectures.

Key Words: Axial symmetry • Elliptical symmetry • z transformation • Zeta transformation


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