Exponential transformation models

Abstract

The class of exponential transformation models, i. e. transformation models that are also exponential families, is investigated. It is shown, by using standard exponential family theory, that the group of transformations on the sample space generating the model induces affine transformation groups on the parameter space and on the range space of the minimal sufficient statistic, the induced groups being in fact representations of the original transformation group. The model function may be explicitly expressed in terms of these representations and it possesses a number of important properties. The possibility of extending exponential transformation models to larger (composite) exponential transformation models is considered, such extensions serving, for instance, the purpose of model control. The inference for (composite) exponential transformation models is described in general terms and examples of such models are discussed. The paper starts with an introduction to transformation models and a unified treatment of their distribution theory, required in the subsequent discussion of the exponential transformation models.