nag_rngs_gamma (g05lfc) : NAG Library, Mark 23

nag_rngs_gamma (g05lfc) generates a vector of pseudorandom numbers taken from a gamma distribution with

a

and

b

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 ​ 0 \leq x; a, b > 0.0 \\ 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 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 functions nag_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_gamma (g05lfc).

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

On entry:

a

, the parameter of the gamma distribution.

Constraint:

a > 0.0

On entry:

b

, the parameter of the gamma distribution.

Constraint:

b > 0.0

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 0

On exit: the

n

pseudorandom numbers from the specified gamma distribution.

On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).

On entry: contains values which define the current state of the selected generator.

On exit: contains updated values defining the new state of the selected generator.

The NAG error argument (see Section 3.6 in the Essential Introduction).

On entry, argument

〈value〉

had an illegal value.

On entry,

n = 〈value〉

.
Constraint:

n \geq 0

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.

On entry,

a = 〈value〉

.
Constraint:

a > 0.0

On entry,

b = 〈value〉

.
Constraint:

b > 0.0

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 nag_rngs_gamma (g05lfc), after initialization by nag_rngs_init_repeatable (g05kbc).

NAG Library Function Document

nag_rngs_gamma (g05lfc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 Function Documentnag_rngs_gamma (g05lfc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 Function Document

nag_rngs_gamma (g05lfc)