where G(.,.) is the upper incomplete gamma function
Notations f(*): Probability density function E(*): Mean function Var(*): Variance function [gamma](y, x): Lower incomplete gamma function [GAMMA](y, x): Upper incomplete gamma function
F(*): Cumulative distribution function S(*): Survival function h(*): Hazard rate function L(*): Likelihood function l(*): Log-likelihood function [psi](*): Digamma function [psi]'(*): Trigamma function I(*): Expected Fisher information matrix [mu]: Positive parameter [phi]: Positive parameter [alpha]: Positive parameter [beta]: Real parameter [lambda]: Positive parameter Q(*): Quantile function [THETA]: Vector of parameters H(*: Observed Fisher information matrix [[mu].sub.k]: kth moment G(*: TTT-plot [D.sub.n]: Kolmogorov-Smirnov statistic [y.sup.*]: Predictive value.
where [gamma](a, z) is the lower incomplete gamma function which is related to the gamma function by [gamma](a, z) = [GAMMA](a) - [GAMMA](a, z), where [GAMMA](a, z) is the upper incomplete gamma function
defined by [[integral].sup.[infinity].sub.z] [x.sup.a-1][e.sup.-x]dx.
where [GAMMA](a,z) = [[integral].sup.[infinity].sub.z] [t.sup.a-1][e.sup.-t]dt denotes the upper incomplete gamma function
The term in the denominator is the complete gamma function and term in the numerator is known as the upper incomplete gamma function
. Factoring out Z and also [UB.sub.Sa] requires computing the inverse gamma function.
Denote by [GAMMA](a, z) the upper incomplete gamma function (cf.
In addition, the usual exponential integral function [E.sub.s](z) can be defined in terms of the upper incomplete gamma function as follows (cf.
We highlight that the moment generating function of the Muth distribution can also be expressed alternatively in terms of the upper incomplete gamma function by virtue of Eq.
where [GAMMA](x, x) is the upper incomplete gamma function [19, equation (8.350.2)] and [Q.sub.u](x,x) is the uth order generalized Marcum Q-function defined as 
An alternate canonical series form representation of the generalized Marcum Q-function is reported in [21, equation (4.74)], which can be further simplified by noting the relationship of finite summation series with upper incomplete gamma function as outlined in 19, Section 8.352] to give
As the generalized error function [Erf.sub.a] is defined in (4), through the upper incomplete gamma function [GAMMA]([a.sup.-1], x), series expansions can be used for a more "numerical-oriented" form of (4).
The values of the inverse upper incomplete gamma function [[GAMMA].sup.-1] (([gamma]- 1)/[gamma], *) were numerically calculated.