The Metalog Distributions

SPT Bounded Metalog Equations

The equations for the SPT-parameterized bounded metalog distribution are shown below. These equations are implemented in the SPT Metalogs Excel workbookIn this case, the metalog quantile function and pdf, Mlogit(y) and mlogit(y) respectively, are the n=3 special case of the general bounded metalog equations:
        Mlogit(y)  = (bl+beM(y)) / (1+eM(y))                          for 0 < y < 1
= bl                                                               for y = 0
                      = bu                                                       for y = 1

 mlogit(y)  = m(y(1+eM(y))2/((bu-bl)eM(y))               for 0 < y < 1
                      = 0                                                         for y = 0 or y = 1

where bl and bu are user-specified lower and upper bounds respectively and

M(y) =


m(y) =


The constants

can be calculated directly from SPT input parameters:

The feasibility requirement is given by:


For further information, see The Metalog Distributions, Section 6.2.1.