NAG FL Interface
g01fff (inv_​cdf_​gamma)

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1 Purpose

g01fff returns the deviate associated with the given lower tail probability of the gamma distribution.

2 Specification

Fortran Interface
Function g01fff ( p, a, b, tol, ifail)
Real (Kind=nag_wp) :: g01fff
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: p, a, b, tol
C Header Interface
#include <nag.h>
double  g01fff_ (const double *p, const double *a, const double *b, const double *tol, Integer *ifail)
The routine may be called by the names g01fff or nagf_stat_inv_cdf_gamma.

3 Description

The deviate, gp, associated with the lower tail probability, p, of the gamma distribution with shape parameter α and scale parameter β, is defined as the solution to
P(Ggp:α,β)=p=1βαΓ(α) 0gpe-G/βGα-1dG,  0gp<;α,β>0.  
The method used is described by Best and Roberts (1975) making use of the relationship between the gamma distribution and the χ2-distribution.
Let y=2 gpβ . The required y is found from the Taylor series expansion
y=y0+rCr(y0) r! (Eϕ(y0) ) r,  
where y0 is a starting approximation
For most values of p and α the starting value
y01=2α (z19α +1-19α ) 3  
is used, where z is the deviate associated with a lower tail probability of p for the standard Normal distribution.
For p close to zero,
y02= (pα2αΓ(α)) 1/α  
is used.
For large p values, when y01>4.4α+6.0,
is found to be a better starting value than y01.
For small α (α0.16), p is expressed in terms of an approximation to the exponential integral and y04 is found by Newton–Raphson iterations.
Seven terms of the Taylor series are used to refine the starting approximation, repeating the process if necessary until the required accuracy is obtained.

4 References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the χ2 distribution Appl. Statist. 24 385–388

5 Arguments

1: p Real (Kind=nag_wp) Input
On entry: p, the lower tail probability from the required gamma distribution.
Constraint: 0.0p<1.0.
2: a Real (Kind=nag_wp) Input
On entry: α, the shape parameter of the gamma distribution.
Constraint: 0.0<a106.
3: b Real (Kind=nag_wp) Input
On entry: β, the scale parameter of the gamma distribution.
Constraint: b>0.0.
4: tol Real (Kind=nag_wp) Input
On entry: the relative accuracy required by you in the results. The smallest recommended value is 50×δ, where δ=max(10-18,machine precision). If g01fff is entered with tol less than 50×δ or greater or equal to 1.0, then 50×δ is used instead.
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 -1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. 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:
Note: in some cases g01fff may return useful information.
If on exit ifail=1, 2, 3 or 5, then g01fff returns 0.0.
On entry, p=value.
Constraint: p<1.0.
On entry, p=value.
Constraint: p0.0.
On entry, a=value.
Constraint: a>0.0.
On entry, a=value.
Constraint: a106.
On entry, b=value.
Constraint: b>0.0.
The probability is too close to 0.0 for the given a to enable the result to be calculated.
The algorithm has failed to converge in 100 iterations. A larger value of tol should be tried. The result may be a reasonable approximation.
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
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

In most cases the relative accuracy of the results should be as specified by tol. However, for very small values of α or very small values of p there may be some loss of accuracy.

8 Parallelism and Performance

g01fff is not threaded in any implementation.

9 Further Comments


10 Example

This example reads lower tail probabilities for several gamma distributions, and calculates and prints the corresponding deviates until the end of data is reached.

10.1 Program Text

Program Text (g01fffe.f90)

10.2 Program Data

Program Data (g01fffe.d)

10.3 Program Results

Program Results (g01fffe.r)