nag_deviates_gamma_dist (g01ffc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_deviates_gamma_dist (g01ffc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagg01.h>
double  nag_deviates_gamma_dist (double p, double a, double b, double tol, NagError *fail)

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
PGgp:α,β=p=1βαΓα 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+rCry0 r! Eϕy0 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,
y03=-2ln1-p-α-1ln12y01+lnΓ α
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:     pdoubleInput
On entry: p, the lower tail probability from the required gamma distribution.
Constraint: 0.0p<1.0.
2:     adoubleInput
On entry: α, the shape parameter of the gamma distribution.
Constraint: 0.0<a106.
3:     bdoubleInput
On entry: β, the scale parameter of the gamma distribution.
Constraint: b>0.0.
4:     toldoubleInput
On entry: the relative accuracy required by you in the results. The smallest recommended value is 50×δ, where δ=max10-18,machine precision. If nag_deviates_gamma_dist (g01ffc) is entered with tol less than 50×δ or greater or equal to 1.0, then 50×δ is used instead.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On any of the error conditions listed below, except NE_ALG_NOT_CONV, nag_deviates_gamma_dist (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.
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.
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  Further Comments


9  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.

9.1  Program Text

Program Text (g01ffce.c)

9.2  Program Data

Program Data (g01ffce.d)

9.3  Program Results

Program Results (g01ffce.r)

nag_deviates_gamma_dist (g01ffc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012