Oracle estimation of parametric transformation models

Transformation models, like the Box-Cox transformation, are widely used in regression to reduce non-additivity, non-normality, and heteroscedasticity. The question of whether one may or may not treat the estimated transformation parameter as fixed in inference about other model parameters has a long and controversial history (Bickel and Doksum, 1981, Hinkley and Runger, 1984). While the frequentist wisdom is that uncertainty regarding the true value of the transformation parameter cannot be ignored, in practice, difficulties in interpretation arise if the transformation is regarded as random and not fixed. In this paper, we suggest a golden mean methodology which attempts to reconcile these philosophies. Penalized estimation yields oracle estimates of transformations that enable treating the transformation parameter as known when the data indicate one of a prespecified set of transformations of scientific interest. When the true transformation is outside this set, rigorous frequentist inference is still achieved. The methodology permits multiple candidate values for the transformation, as is common in applications, as well as simultaneously accommodating variable selection in regression model. Theoretical issues, such as selection consistency and the oracle property, are rigorously established. Numerical studies, including extensive simulation studies and real data examples, illustrate the practical utility of the proposed methods.