NAG FL Interface
g01eff (prob_​gamma)

1 Purpose

g01eff returns the lower or upper tail probability of the gamma distribution, with parameters α and β.

2 Specification

Fortran Interface
Function g01eff ( tail, g, a, b, ifail)
Real (Kind=nag_wp) :: g01eff
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: g, a, b
Character (1), Intent (In) :: tail
C Header Interface
#include <nag.h>
double  g01eff_ (const char *tail, const double *g, const double *a, const double *b, Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01eff or nagf_stat_prob_gamma.

3 Description

The lower tail probability for the gamma distribution with parameters α and β, PGg, is defined by:
P Gg ; α,β = 1 βα Γα 0g Gα-1 e-G/β dG ,   α>0.0 , ​ β>0.0 .  
The mean of the distribution is αβ and its variance is αβ2. The transformation Z=Gβ is applied to yield the following incomplete gamma function in normalized form,
P Gg ; α ,β = P Zg/β : α,1.0 = 1 Γα 0g/β Zα-1 e-Z dZ .  
This is then evaluated using s14baf.

4 References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

5 Arguments

1: tail Character(1) Input
On entry: indicates whether an upper or lower tail probability is required.
The lower tail probability is returned, that is PGg:α,β.
The upper tail probability is returned, that is PGg:α,β.
Constraint: tail='L' or 'U'.
2: g Real (Kind=nag_wp) Input
On entry: g, the value of the gamma variate.
Constraint: g0.0.
3: a Real (Kind=nag_wp) Input
On entry: the parameter α of the gamma distribution.
Constraint: a>0.0.
4: b Real (Kind=nag_wp) Input
On entry: the parameter β of the gamma distribution.
Constraint: b>0.0.
5: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
If ifail=1, 2, 3 or 4 on exit, then g01eff returns 0.0.
On entry, tail=value.
Constraint: tail='L' or 'U'.
On entry, g=value.
Constraint: g0.0.
On entry, a=value and b=value.
Constraint: a>0.0 and b>0.0.
The algorithm has failed to converge in value iterations. The probability returned should be a reasonable approximation to the solution.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The result should have a relative accuracy of machine precision. There are rare occasions when the relative accuracy attained is somewhat less than machine precision but the error should not exceed more than 1 or 2 decimal places. Note also that there is a limit of 18 decimal places on the achievable accuracy, because constants in s14baf are given to this precision.

8 Parallelism and Performance

g01eff is not threaded in any implementation.

9 Further Comments

The time taken by g01eff varies slightly with the input arguments g, a and b.

10 Example

This example reads in values from a number of gamma distributions and computes the associated lower tail probabilities.

10.1 Program Text

Program Text (g01effe.f90)

10.2 Program Data

Program Data (g01effe.d)

10.3 Program Results

Program Results (g01effe.r)