The gamma distribution is used frequently in the statistical analysis of hydrologic data. For example, Pearson type III and log-Pearson type III distributions used in the flood frequency analysis are members of the gamma distribution family. It is a very versatile distribution the PDF of which can take many forms (see Fig. 2.20). The PDF of a two-parameter gamma random variable, denoted by X ~ GAM(a, в), is given by Eq. (2.72). The standard gamma PDF involving one-parameter a can be derived using variable transformation by letting
Y = X/в. The PDF of the standard gamma random variable Y, denoted by
Y ~ GAM(a), is shown in Eq. (2.78). The standard gamma distribution is used in all algorithms to generate gamma random variate Y s from which random variates from a two-parameter gamma distribution are obtained from X = вY.
The simplest case in generating gamma random variates is when the shape parameter a is a positive integer (Erlang distribution). In such a case, the random variable Y ~ GAM(a) is a sum of a independent and identical standard exponential random variables with parameter в = 1. The random variates from
Y ~ GAM(a), then, can be obtained as
a
Y = £- ln(U) (6.21)
i = 1
To avoid large numbers oflogarithmic evaluations (when a is large), Eq. (6.21) alternatively can be expressed as
Y = -1п^П Uij (6.22)
Although simplicity is the idea, this algorithm for generating gamma random variates has three disadvantages: (1) It is only applicable to integer-valued shape parameter a, (2) the algorithm becomes extremely slow when a is large, and (3) for a large a, numerical underflow on a computer could occur.
Several algorithms have been developed for generating standard gamma random variates for a real-valued a. The algorithms can be classified into those which are applicable for the full range (a > 0), 0 < a < 1, and a > 1. Dagpunar (1988) showed that through a numerical experiment, algorithms developed for a full range of a are not efficient in comparison with those especially tailored for subregions. The two efficient AR-based algorithms are presented in Dagpunar (1988).
