NAG Library Function Document

nag_random_gamma (g05ffc)

the gamma distribution reduces to the exponential distribution and the method based on the logarithmic transformation of a uniform random variate is used.

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.

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: a – doubleInput: On entry: the argument, $a$ , of the gamma distribution.
Constraint: $a > 0.0$ .
2: b – doubleInput: On entry: the argument, $b$ , of the gamma distribution.
Constraint: $b > 0.0$ .
3: n – IntegerInput: On entry: the number, $n$ , of pseudorandom numbers to be generated.
Constraint: $n \geq 1$ .
4: x[n] – doubleOutput: On exit: the $n$ pseudorandom variates from the specified gamma distribution.
5: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_INT_ARG_LE: On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .
NE_REAL_ARG_LE: On entry, a must not be less than or equal to 0.0: $a = 〈value〉$ .; On entry, b must not be less than or equal to 0.0: $b = 〈value〉$ .

7 Accuracy

Not applicable.

8 Further Comments

To generate an observation from the

χ^{2}

distribution with

v

degrees of freedom generate an observation from a gamma distribution with arguments

a = v / 2

b = 2

To generate an observation,

y

, from a Student's

t

-distribution with degrees of freedom

v

generate an observation,

x

, from a gamma distribution with arguments

a = v / 2

and

b = 2

and an observation,

z

, from a standard Normal distribution (see nag_random_normal (g05ddc)) and use the transformation

y = z / \sqrt{x}

9 Example

The example program prints a set of five pseudorandom variates from a gamma distribution with arguments

a = 5.0

and

b = 1.0

, generated by nag_random_gamma (g05ffc) after initialization by nag_random_init_repeatable (g05cbc).

9.1 Program Text

Program Text (g05ffce.c)

9.2 Program Data

None.

9.3 Program Results

Program Results (g05ffce.r)

nag_random_gamma (g05ffc) (PDF version)

g05 Chapter Contents

g05 Chapter Introduction

NAG C Library Manual

NAG Library Function Documentnag_random_gamma (g05ffc)

+− Contents

1 Purpose

2 Specification

3 Description