nag_deviates_gamma_vector (g01tfc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_deviates_gamma_vector (g01tfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_deviates_gamma_vector (g01tfc) returns a number of deviates associated with given probabilities of the gamma distribution.

2  Specification

#include <nag.h>
#include <nagg01.h>
void  nag_deviates_gamma_vector (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer la, const double a[], Integer lb, const double b[], double tol, double g[], Integer ivalid[], NagError *fail)

3  Description

The deviate, gpi, associated with the lower tail probability, pi, of the gamma distribution with shape parameter αi and scale parameter βi, is defined as the solution to
P Gi gpi :αi,βi = pi = 1 βi αi Γ αi 0 gpi ei - Gi / βi Gi αi-1 dGi ,   0 gpi < ; ​ αi , βi > 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 yi=2 gpiβi . The required yi is found from the Taylor series expansion
yi=y0+rCry0 r! Eiϕy0 r,
where y0 is a starting approximation
For most values of pi and αi the starting value
y01=2αi zi19αi +1-19αi 3
is used, where zi is the deviate associated with a lower tail probability of pi for the standard Normal distribution.
For pi close to zero,
y02= piαi2αiΓ αi 1/αi
is used.
For large pi values, when y01>4.4αi+6.0,
y03=-2ln1-pi-αi-1ln12y01+lnΓ αi
is found to be a better starting value than y01.
For small αi αi0.16, pi 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.
The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the g01 Chapter Introduction for further information.

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:     ltailIntegerInput
On entry: the length of the array tail.
Constraint: ltail>0.
2:     tail[ltail]const Nag_TailProbabilityInput
On entry: indicates which tail the supplied probabilities represent. For j= i-1 mod ltail , for i=1,2,,maxltail,lp,la,lb:
tail[j]=Nag_LowerTail
The lower tail probability, i.e., pi = P Gi gpi : αi , βi .
tail[j]=Nag_UpperTail
The upper tail probability, i.e., pi = P Gi gpi : αi , βi .
Constraint: tail[j-1]=Nag_LowerTail or Nag_UpperTail, for j=1,2,,ltail.
3:     lpIntegerInput
On entry: the length of the array p.
Constraint: lp>0.
4:     p[lp]const doubleInput
On entry: pi, the probability of the required gamma distribution as defined by tail with pi=p[j], j=i-1 mod lp.
Constraints:
  • if tail[k]=Nag_LowerTail, 0.0p[j]<1.0;
  • otherwise 0.0<p[j]1.0.
Where k=i-1 mod ltail and j=i-1 mod lp.
5:     laIntegerInput
On entry: the length of the array a.
Constraint: la>0.
6:     a[la]const doubleInput
On entry: αi, the first parameter of the required gamma distribution with αi=a[j], j=i-1 mod la.
Constraint: 0.0<a[j-1]106, for j=1,2,,la.
7:     lbIntegerInput
On entry: the length of the array b.
Constraint: lb>0.
8:     b[lb]const doubleInput
On entry: βi, the second parameter of the required gamma distribution with βi=b[j], j=i-1 mod lb.
Constraint: b[j-1]>0.0, for j=1,2,,lb.
9:     toldoubleInput
On entry: the relative accuracy required by you in the results. If nag_deviates_gamma_vector (g01tfc) is entered with tol greater than or equal to 1.0 or less than 10×machine precision (see nag_machine_precision (X02AJC)), then the value of 10×machine precision is used instead.
10:   g[dim]doubleOutput
Note: the dimension, dim, of the array g must be at least maxltail,lp,la,lb.
On exit: gpi, the deviates for the gamma distribution.
11:   ivalid[dim]IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least maxltail,lp,la,lb.
On exit: ivalid[i-1] indicates any errors with the input arguments, with
ivalid[i-1]=0
No error.
ivalid[i-1]=1
On entry,invalid value supplied in tail when calculating gpi.
ivalid[i-1]=2
On entry,invalid value for pi.
ivalid[i-1]=3
On entry,αi0.0,
orαi>106,
orβi0.0.
ivalid[i-1]=4
pi is too close to 0.0 or 1.0 to enable the result to be calculated.
ivalid[i-1]=5
The solution has failed to converge. The result may be a reasonable approximation.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_ARRAY_SIZE
On entry, array size=value.
Constraint: la>0.
On entry, array size=value.
Constraint: lb>0.
On entry, array size=value.
Constraint: lp>0.
On entry, array size=value.
Constraint: ltail>0.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INTERNAL_ERROR
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.
NW_IVALID
On entry, at least one value of tail, p, a, or b was invalid.
Check ivalid for more information.

7  Accuracy

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

8  Further Comments

None.

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 (g01tfce.c)

9.2  Program Data

Program Data (g01tfce.d)

9.3  Program Results

Program Results (g01tfce.r)


nag_deviates_gamma_vector (g01tfc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

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