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.
- Two-piece distribution
Jones, M. C., & Anaya-Izquierdo, K. (2011). On parameter orthogonality in symmetric and skew models. Journal of Statistical Planning and Inference, 141(2), 758 - 770. https://doi.org/10.1016/j.jspi.2010.07.021