g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rand_gamma (g05sjc)

## 1  Purpose

nag_rand_gamma (g05sjc) generates a vector of pseudorandom numbers taken from a gamma distribution with parameters $a$ and $b$.

## 2  Specification

 #include #include
 void nag_rand_gamma (Integer n, double a, double b, Integer state[], double x[], NagError *fail)

## 3  Description

The gamma distribution has PDF (probability density function)
 $fx= 1baΓa xa-1e-x/b if ​x≤0; a,b>0 fx=0 otherwise.$
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 ${f}_{1}\left(x\right)=ca{x}^{a-1}/{t}^{a}$ and ${f}_{2}\left(x\right)=\left(1-c\right){e}^{-\left(x-t\right)}$, where $c=t/\left(t+a{e}^{-t}\right)$, and the switching argument, $t$, 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 functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_gamma (g05sjc).

## 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:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
On entry: $a$, the parameter of the gamma distribution.
Constraint: ${\mathbf{a}}>0.0$.
3:     bdoubleInput
On entry: $b$, the parameter of the gamma distribution.
Constraint: ${\mathbf{b}}>0.0$.
4:     state[$\mathit{dim}$]IntegerCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization functions nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
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]doubleOutput
On exit: the $n$ pseudorandom numbers from the specified gamma distribution.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
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.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_REAL
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}>0.0$.

Not applicable.

None.

## 9  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 nag_rand_gamma (g05sjc), after initialization by nag_rand_init_repeatable (g05kfc).

### 9.1  Program Text

Program Text (g05sjce.c)

None.

### 9.3  Program Results

Program Results (g05sjce.r)