The Metalog Distributions

SPT Semi-Bounded Metalog Equations
The equations for the SPT-parameterized semi-bounded metalog distribution are shown below. These equations are implemented in the SPT Metalogs Excel workbook.In this case, the metalog quantile function and pdf, Mlog(y) and mlog(y) respectively, are the n=3 special case of the general semi-bounded metalog equations:
 
     Mlog(y)  = bl + eM(y)                     for 0 < y < 1
                   = bl                                  for y = 0

      mlog(y) = m(y) e-M(y)                   for 0 < y < 1
 
                 = 0                               for y = 0

where bl is a user-specified lower bound and

M(y) =
 


m(y) =
 

The constants

can be calculated directly from SPT input parameters:
The feasibility requirement is given by:
where
For further information, see The Metalog Distributions, Section 6.2.1.