G01SFF (PDF version)
G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

G01SFF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

G01SFF returns a number of lower or upper tail probabilities for the gamma distribution.

2  Specification

SUBROUTINE G01SFF ( LTAIL, TAIL, LG, G, LA, A, LB, B, P, IVALID, IFAIL)
INTEGER  LTAIL, LG, LA, LB, IVALID(*), IFAIL
REAL (KIND=nag_wp)  G(LG), A(LA), B(LB), P(*)
CHARACTER(1)  TAIL(LTAIL)

3  Description

The lower tail probability for the gamma distribution with parameters αi and βi, PGigi, is defined by:
P Gi gi :αi,βi = 1 βi αi Γ αi 0 gi Gi αi-1 e -Gi/βi dGi ,   αi>0.0 , ​ βi>0.0 .
The mean of the distribution is αiβi and its variance is αiβi2. The transformation Zi=Giβi is applied to yield the following incomplete gamma function in normalized form,
P Gi gi :αi,βi = P Zi gi / βi :αi,1.0 = 1 Γ αi 0 gi / βi Zi αi-1 e -Zi dZi .
This is then evaluated using S14BAF.
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector parameters 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

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

5  Parameters

1:     LTAIL – INTEGERInput
On entry: the length of the array TAIL.
Constraint: LTAIL>0.
2:     TAIL(LTAIL) – CHARACTER(1) arrayInput
On entry: indicates whether a lower or upper tail probability is required. For j= i-1 mod LTAIL +1 , for i=1,2,,maxLTAIL,LG,LA,LB:
TAILj='L'
The lower tail probability is returned, i.e., pi = P Gi gi :αi,βi .
TAILj='U'
The upper tail probability is returned, i.e., pi = P Gi gi :αi,βi .
Constraint: TAILj='L' or 'U', for j=1,2,,LTAIL.
3:     LG – INTEGERInput
On entry: the length of the array G.
Constraint: LG>0.
4:     G(LG) – REAL (KIND=nag_wp) arrayInput
On entry: gi, the value of the gamma variate with gi=Gj, j=i-1 mod LG+1.
Constraint: Gj0.0, for j=1,2,,LG.
5:     LA – INTEGERInput
On entry: the length of the array A.
Constraint: LA>0.
6:     A(LA) – REAL (KIND=nag_wp) arrayInput
On entry: the parameter αi of the gamma distribution with αi=Aj, j=i-1 mod LA+1.
Constraint: Aj>0.0, for j=1,2,,LA.
7:     LB – INTEGERInput
On entry: the length of the array B.
Constraint: LB>0.
8:     B(LB) – REAL (KIND=nag_wp) arrayInput
On entry: the parameter βi of the gamma distribution with βi=Bj, j=i-1 mod LB+1.
Constraint: Bj>0.0, for j=1,2,,LB.
9:     P(*) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array P must be at least maxLG,LA,LB,LTAIL.
On exit: pi, the probabilities of the beta distribution.
10:   IVALID(*) – INTEGER arrayOutput
Note: the dimension of the array IVALID must be at least maxLG,LA,LB,LTAIL.
On exit: IVALIDi indicates any errors with the input arguments, with
IVALIDi=0
No error.
IVALIDi=1
On entry,invalid value supplied in TAIL when calculating pi.
IVALIDi=2
On entry,gi<0.0.
IVALIDi=3
On entry,αi0.0,
orβi0.0.
IVALIDi=4
The solution did not converge in 600 iterations, see S14BAF for details. The probability returned should be a reasonable approximation to the solution.
11:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. 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:
IFAIL=1
On entry, at least one value of G, A, B or TAIL was invalid, or the solution did not converge.
Check IVALID for more information.
IFAIL=2
On entry, array size=value.
Constraint: LTAIL>0.
IFAIL=3
On entry, array size=value.
Constraint: LG>0.
IFAIL=4
On entry, array size=value.
Constraint: LA>0.
IFAIL=5
On entry, array size=value.
Constraint: LB>0.
IFAIL=-999
Dynamic memory allocation failed.

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.

8  Further Comments

The time taken by G01SFF to calculate each probability varies slightly with the input parameters gi, αi and βi.

9  Example

This example reads in values from a number of gamma distributions and computes the associated lower tail probabilities.

9.1  Program Text

Program Text (g01sffe.f90)

9.2  Program Data

Program Data (g01sffe.d)

9.3  Program Results

Program Results (g01sffe.r)


G01SFF (PDF version)
G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

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