Special functions: Incomplete gamma functions

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The terminology used in literature for the incomplete gamma functions, also called regularized or normalized can be deceiving. In fact there are four "variations" of the function shown in the table below which brings some surprises when comparing results obtained from different applications. For example, when translating code from FORTRAN to BASIC it is quite common to find the former returning upper tails.

In this blog the references to the incomplete gamma function mean the incomplete gamma ratio function.

Excel Gamma distribution pdf

NOTE: It can be found, for example in SPSS manual, the expression of the  Gamma distribution with the parameter β on the numerator. Of course the results obtained are the same being the value of the parameter the inverse between the two expressions. The problem is to use one of the formulas with the β value appropriate for the other.

The incomplete gamma gamma ratio function can be calculated as a particular case of the Gamma cdf distribution (β=1).  We consider only the incomplete gama ratio lower tail since the other can be easily computed from it.  

Excel formulas and user defined functions

' Incomplete gamma ratio function Function xlGAMMAINC(x, a)   With WorksheetFunction     xlGAMMAINC = .Gamma_Dist(x, a, 1, 1)   End With End Function

Matlab

>> gammainc(0.5,0.2) ans =     0.8788  >> gammainc(0.5,0.2,'upper') ans =     0.1212 >> gammainc(0.5,0.2)+gammainc(0.5,0.2,'upper') ans =      1 >> gammainc(0.5,0.2)*gamma(0.2) ans =     4.0343