Distributions Generated by Transformation of Scale Using an Extended Schlömilch Transformation

Baker (2008) shows how more flexible densities on R+ can be generated from others by applying the Schlömilch transformation to the abscissa. Such “transformation of scale” is not even guaranteed to provide integrable functions in general. The appeal of the Schlömilch transformation is that it automatically does so; moreover, the normalising constant is unaffected and hence immediately available. In this paper, we fit the original Schlömilch transformation into a broader framework of novel extended Schlömilch transformations based on self-inverse functions, and propose the corresponding newly generated densities which also retain the same normalising constant. As well as providing parallels with, and extensions of, the many properties of the new densities developed by Baker, we investigate the skewness properties of both original and extended Schlömilch-based distributions via application of a recently proposed density-based approach to quantifying asymmetry. AMS (2000) subject classification. Primary 62E10, secondary 60E99.