NAG FL Interface
g01eff (prob_​gamma)

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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, P(Gg), is defined by:
P ( Gg ; α,β) = 1 βα Γ(α) 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 Γ(α) 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.
tail='L'
The lower tail probability is returned, that is P(Gg:α,β).
tail='U'
The upper tail probability is returned, that is P(Gg: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 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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.
ifail=1
On entry, tail=value.
Constraint: tail='L' or 'U'.
ifail=2
On entry, g=value.
Constraint: g0.0.
ifail=3
On entry, a=value and b=value.
Constraint: a>0.0 and b>0.0.
ifail=4
The algorithm has failed to converge in value iterations. The probability returned should be a reasonable approximation to the solution.
ifail=-99
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.
ifail=-399
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.
ifail=-999
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)