G01KKF (PDF version)
G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


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

G01KKF returns a number of values of the probability density function (PDF), or its logarithm, for the gamma distribution.

2  Specification

REAL (KIND=nag_wp)  X(LX), A(LA), B(LB), PDF(*)

3  Description

The gamma distribution with shape parameter αi and scale parameter βi has PDF
f xi,αi,βi = 1 βi αi Γαi xi αi-1 e -xi / βi if ​ xi 0 ;   αi , βi > 0 fxi,αi,βi=0 otherwise.
If 0.01xi,αi,βi100 then an algorithm based directly on the gamma distribution's PDF is used. For values outside this range, the function is calculated via the Poisson distribution's PDF as described in Loader (2000) (see Section 8).
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

Loader C (2000) Fast and accurate computation of binomial probabilities (not yet published)

5  Parameters

1:     ILOG – INTEGERInput
On entry: the value of ILOG determines whether the logarithmic value is returned in PDF.
fxi,αi,βi, the probability density function is returned.
logfxi,αi,βi, the logarithm of the probability density function is returned.
Constraint: ILOG=0 or 1.
2:     LX – INTEGERInput
On entry: the length of the array X.
Constraint: LX>0.
3:     X(LX) – REAL (KIND=nag_wp) arrayInput
On entry: xi, the values at which the PDF is to be evaluated with xi=Xj, j=i-1 mod LX+1, for i=1,2,,maxLX,LA,LB.
4:     LA – INTEGERInput
On entry: the length of the array A.
Constraint: LA>0.
5:     A(LA) – REAL (KIND=nag_wp) arrayInput
On entry: αi, the shape parameter with αi=Aj, j=i-1 mod LA+1.
Constraint: Aj>0.0, for j=1,2,,LA.
6:     LB – INTEGERInput
On entry: the length of the array B.
Constraint: LB>0.
7:     B(LB) – REAL (KIND=nag_wp) arrayInput
On entry: βi, the scale parameter with βi=Bj, j=i-1 mod LB+1.
Constraint: Bj>0.0, for j=1,2,,LB.
8:     PDF(*) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array PDF must be at least maxLX,LA,LB.
On exit: fxi,αi,βi or logfxi,αi,βi.
9:     IVALID(*) – INTEGER arrayOutput
Note: the dimension of the array IVALID must be at least maxLX,LA,LB.
On exit: IVALIDi indicates any errors with the input arguments, with
No error.
xiβi overflows, the value returned should be a reasonable approximation.
10:   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:
On entry, at least one value of X, A or B was invalid.
Check IVALID for more information.
On entry, ILOG=value.
Constraint: ILOG=0 or 1.
On entry, array size=value.
Constraint: LX>0.
On entry, array size=value.
Constraint: LA>0.
On entry, array size=value.
Constraint: LB>0.

7  Accuracy

Not applicable.

8  Further Comments

Due to the lack of a stable link to Loader (2000) paper, we give a brief overview of the method, as applied to the Poisson distribution. The Poisson distribution has a continuous mass function given by,
px;λ = λx x! e-λ . (1)
The usual way of computing this quantity would be to take the logarithm and calculate,
log p x;λ = x logλ - log x! - λ .
For large x and λ, xlogλ and logx! are very large, of the same order of magnitude and when calculated have rounding errors. The subtraction of these two terms can therefore result in a number, many orders of magnitude smaller and hence we lose accuracy due to subtraction errors. For example for x=2×106 and λ=2×106, logx!2.7×107 and logpx;λ=-8.17326744645834. But calculated with the method shown later we have logpx;λ=-8.1732674441334492. The difference between these two results suggests a loss of about 7 significant figures of precision.
Loader introduces an alternative way of expressing (1) based on the saddle point expansion,
log p x;λ = log p x;x - Dx;λ , (2)
where Dx;λ, the deviance for the Poisson distribution is given by,
Dx;λ = log p x;x - log p x;λ , = λ D0 x λ , (3)
D0 ε = ε logε + 1 - ε .
For ε close to 1, D0ε can be evaluated through the series expansion
λ D0 x λ = x-λ 2 x+λ + 2x j=1 v 2j+1 2j+1 ,  where ​ v = x-λ x+λ ,
otherwise D0ε can be evaluated directly. In addition, Loader suggests evaluating logx! using the Stirling–De Moivre series,
logx! = 12 log 2πx + x logx -x + δx , (4)
where the error δx is given by
δx = 112x - 1 360x3 + 1 1260x5 + O x-7 .
Finally logpx;λ can be evaluated by combining equations (1)(4) to get,
p x;λ = 1 2πx e - δx - λ D0 x/λ .

9  Example

This example prints the value of the gamma distribution PDF at six different points xi with differing αi and βi.

9.1  Program Text

Program Text (g01kkfe.f90)

9.2  Program Data

Program Data (g01kkfe.d)

9.3  Program Results

Program Results (g01kkfe.r)

Produced by GNUPLOT 4.4 patchlevel 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 9 10 y x Example Program Plots of the Gamma Distribution a=2, b=2 a=9, b=0.5

G01KKF (PDF version)
G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

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