On parameter orthogonality in symmetric and skew models

Orthogonal and partly orthogonal reparametrisations are provided for certain wide and important families of univariate continuous distributions. First, the orthogonality of parameters in location-scale symmetric families is extended to symmetric distributions involving a third parameter. This sets the scene for consideration of the four-parameter situation in which skewness is also allowed. It turns out that one specific approach to generating such four-parameter families, that of two-piece distributions with a certain parametrisation restriction, has some attractive features with regard to parameter orthogonality which, to the best of our knowledge, are not shared with other four-parameter distributions. Our work also affords partly orthogonal parametrisations of three-parameter two-piece models.