Scale equivariance and the Box-Cox transformation

The Box-Cox family of transformations has two useful features: first, it includes linear and logarithmic transformations as special cases; and, second, it possesses strong scale equivariance properties, including the property that the transformation parameter is unaffected by the rescaling. Its main disadvantage is that both the domain and the range of the transformation are, in general, bounded. We show that, for a certain class of models, if a model demonstrates these scale equivariance properties for the dependent variable then the transformation on the dependent variable must be a variant of the Box-Cox transformation.