# Computing Complete Gamma Function

## Problem

Can SPSS/PASW Statistics compute the complete gamma function?

## Resolving The Problem

SPSS 6.0 and above can compute the gamma function for positive values of the variable A. The CDF.GAMMA(X,A,1) function, which calculates the cumulative distribution function for variable X under a gamma distribution with shape=A and scale=1, is an incomplete gamma function ratio (or simply an incomplete gamma function) (see Johnson, Kotz, & Balakrishnan, "Continuous Univariate Distributions", 2nd Ed., 1994, Chap. 3). You can use a recursive relationship between the incomplete gamma functions for A and A+1, i.e. Gi(X,A) and Gi(X,A+1), to find the complete gamma function for A, Gc(A). The recursive relationship is

Gi(X,A) = X**A * EXP[-X]/Gc(A+1) + Gi(X,A+1).

Now set X to 1, giving Gi(1,A) = EXP[-1]/Gc(A+1) + Gi(1,A+1).

which gives Gc(A+1) = EXP[-1]/(Gi(1,A) - Gi(1,A+1)).

Since Gc(A) = Gc(A+1)/A for A>0

Gc(A) = EXP[-1]/A/(Gi(1,A) - Gi(1,A+1)).

The SPSS command to implement this assignment is

COMPUTE GAMA = EXP(-1)/A/(CDF.GAMMA(1,A,1) - CDF.GAMMA(1,A+1,1)).

where A is a variable in the SPSS active file. GAMA will be defined for positive values of A

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Modified date:
16 June 2018

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