A new parameterization of Johnsons SB distribution with application to fitting forest tree diameter data

The SB distributional model of Johnsonʹs 1949 paper was introduced by a transformation to normality, that is, z ~ N(0, 1), consisting of a linear scaling to the range (0, 1), a logit transformation, and an affine transformation, z = (gamma) + (delta)u. The model, in its original parameterization, has often been used in forest diameter distribution modelling. In this paper, we define the SB distribution in terms of the inverse transformation from normality, including an initial linear scaling transformation, u = (gamma)ʹ + (delta)ʹz ((delta)ʹ = 1/(delta) and (gamma)ʹ = -(gamma)/(delta)). The SB model in terms of the new parameterization is derived, and maximum likelihood estimation schema are presented for both model parameterizations. The statistical properties of the two alternative parameterizations are compared empirically on 20 data sets of diameter distributions of Changbai larch (Larix olgensis Henry). The new parameterization is shown to be statistically better than Johnsonʹs original parameterization for the data sets considered here.