G05SJF : NAG Library, Mark 24

3 Description

The gamma distribution has PDF (probability density function)

\begin{array}{l} f (x) = \frac{1}{b^{a} Γ (a)} x^{a - 1} e^{- x / b} & if ​ x \leq 0; a, b > 0 \\ f (x) = 0 & otherwise. \end{array}

One of three algorithms is used to generate the variates depending upon the value of

a

(i)	if $a < 1$ , a switching algorithm described by Dagpunar (1988) (called G6) is used. The target distributions are $f_{1} (x) = c a x^{a - 1} / t^{a}$ and $f_{2} (x) = (1 - c) e^{- (x - t)}$ , where $c = t / (t + a e^{- t})$ , and the switching parameter, $t$ , is taken as $1 - a$ . This is similar to Ahrens and Dieter's GS algorithm (see Ahrens and Dieter (1974)) in which $t = 1$ ;
(ii)	if $a = 1$ , the gamma distribution reduces to the exponential distribution and the method based on the logarithmic transformation of a uniform random variate is used;
(iii)	if $a > 1$ , the algorithm given by Best (1978) is used. This is based on using a Student's $t$ -distribution with two degrees of freedom as the target distribution in an envelope rejection method.

One of the initialization routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05SJF.

4 References

Ahrens J H and Dieter U (1974) Computer methods for sampling from gamma, beta, Poisson and binomial distributions Computing 12 223–46

Best D J (1978) Letter to the Editor Appl. Statist. 27 181

Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press

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

5 Parameters

1: N – INTEGERInput

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

N \geq 0

2: A – REAL (KIND=nag_wp)Input

On entry:

a

, the parameter of the gamma distribution.

Constraint:

A > 0.0

3: B – REAL (KIND=nag_wp)Input

On entry:

b

, the parameter of the gamma distribution.

Constraint:

B > 0.0

4: STATE(

*

) – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

5: X(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the

n

pseudorandom numbers from the specified gamma distribution.

6: 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

- 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,

N < 0

IFAIL = 2

On entry,

A \leq 0.0

IFAIL = 3

On entry,

B \leq 0.0

IFAIL = 4

On entry,

STATE vector was not initialized or has been corrupted.

NAG Library Routine Document

G05SJF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG05SJF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G05SJF