(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 Arguments

1: $n$ – Integer Input: 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 array Communication 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) array Output: On exit: the $n$ pseudorandom numbers from the specified gamma distribution.
6: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = 2$: On entry, $a = ⟨ value ⟩$ .
Constraint: $a > 0.0$ .

$ifail = 3$: On entry, $b = ⟨ value ⟩$ .
Constraint: $b > 0.0$ .

$ifail = 4$: On entry, state vector has been corrupted or not initialized.

$ifail = 7$: On entry, $a = ⟨ value ⟩$ .
For very small shape parameter values, variates are approximate.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

For very small values of the shape parameter,

a

, the G6 algorithm is not appropriate due to loss of accuracy in the transformations used; for example,

t = 1 - a

. An alternative algorithm based on approximations to the density functions has been added at MK30.2 for the case

a < \sqrt{ε}

, where

ε

is the machine precision. A warning exit has also been added for this case to indicate that approximations have been used.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g05sjf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example prints a set of five pseudorandom numbers from a gamma distribution with parameters

a = 5.0

and

b = 1.0

, generated by a single call to g05sjf, after initialization by g05kff.

g05sj: FL CL CPP AD PY MB

NAG FL Interfaceg05sjf (dist_​gamma)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g05sjf (dist_gamma)