Van Zwet ordering for Fechner asymmetry

There are several procedures to construct a skewed distribution. One of these procedures splits the value of a parameter of scale for the two halfs of a symmetric distribution. Fechner proposed this procedure in his famous book Kollektivmaßlehre (1897), p. 295ff.. A similar proposal comes from Fernandez et al. (1995). We consider the very general approach from Arellano-Valle et al. (2005) of splitting a scale parameter and show that this technique of generating skewed distributions incorporates a well-defined parameter of skewness. It is well-defined in the sense that the parameter of skewness is compatible with the ordering <2 of van Zwet (1964) which is the strongest ordering in the hierarchy of orderings discussed by Oja (1981). For this family of skewed distributions it will be shown that the measure proposed by Arnold & Groeneveld (1995) is a measure of skewness in the sense of Oja (1981). In the special case considered by Fechner (1897) this measure and the skewness parameter coincide.