NAG CL Interface
g01ffc (inv_​cdf_​gamma)

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1 Purpose

g01ffc returns the deviate associated with the given lower tail probability of the gamma distribution.

2 Specification

#include <nag.h>
double  g01ffc (double p, double a, double b, double tol, NagError *fail)
The function may be called by the names: g01ffc, nag_stat_inv_cdf_gamma or nag_deviates_gamma_dist.

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 double Input
On entry: p, the lower tail probability from the required gamma distribution.
Constraint: 0.0p<1.0.
2: a double Input
On entry: α, the shape parameter of the gamma distribution.
Constraint: 0.0<a106.
3: b double Input
On entry: β, the scale parameter of the gamma distribution.
Constraint: b>0.0.
4: tol double 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 g01ffc is entered with tol less than 50×δ or greater or equal to 1.0, then 50×δ is used instead.
5: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If on exit fail.code= NE_GAM_NOT_CONV, NE_PROBAB_CLOSE_TO_TAIL, NE_REAL_ARG_GE, NE_REAL_ARG_GT, NE_REAL_ARG_LE or NE_REAL_ARG_LT, then g01ffc returns 0.0.
On any of the error conditions listed below, except fail.code= NE_ALG_NOT_CONV, g01ffc returns 0.0 .
The algorithm has failed to converge in 100 iterations. A larger value of tol should be tried. The result may be a reasonable approximation.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL 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 CL Interface for further information.
The probability is too close to 0.0 for the given a to enable the result to be calculated.
On entry, p=value.
Constraint: p<1.0.
On entry, a=value.
Constraint: a106.
On entry, a=value.
Constraint: a>0.0.
On entry, b=value.
Constraint: b>0.0.
On entry, p=value.
Constraint: p0.0.

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

Background information to multithreading can be found in the Multithreading documentation.
g01ffc 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 (g01ffce.c)

10.2 Program Data

Program Data (g01ffce.d)

10.3 Program Results

Program Results (g01ffce.r)