Van Zwet ordering and the Ferreira-Steel family of skewed distributions

There are several procedures to construct a skewed distribution. One of these procedures is based on a symmetric distribution that will be distorted by a skewed distribution defined on (0; 1). This proposal stems from Arellano-Valle et al. and was refined by Ferreira and Steel. Up to now, it is an open question whether the famous skewness ordering of van Zwet will be preserved for this proposal. There is a general condition under which the van Zwets skewness ordering will be preserved by the Ferreira-Steel family. But this condition is not easy to verify for the most families of distribution. Therefore, for the skewness mechanism we choose a special beta distribution with only one parameter. Then, we get three results. First, the skewness ordering will be preserved starting for symmetric distributions that are leptokurtic like the logistic distribution. Larger parameter values give distributions that are more skewed to the right. Second, the same skewness mechanism can generate distributions that are more skewed to left if the support of the underlying symmetric distribution is compact. Third, for underlying symmetric distributions on R with platykurtic behavior the van Zwet ordering of skewness will be preserved. This restricts a little bit the benefit of the Ferreira-Steel family.