naginterfaces.library.rand.dist_​gamma

naginterfaces.library.rand.dist_gamma(n, a, b, statecomm)

dist_gamma generates a vector of pseudorandom numbers taken from a gamma distribution with parameters $a$ and $b$.

For full information please refer to the NAG Library document for g05sj

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g05/g05sjf.html

Parameters

nint

$n$, the number of pseudorandom numbers to be generated.

afloat

$a$, the parameter of the gamma distribution.

bfloat

$b$, the parameter of the gamma distribution.

statecommdict, RNG communication object, modified in place

RNG communication structure.

This argument must have been initialized by a prior call to init_repeat() or init_nonrepeat().

Returns

xfloat, ndarray, shape $\left(n\right)$

The $n$ pseudorandom numbers from the specified gamma distribution.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $2$)

On entry, $a=⟨value⟩$.

Constraint: $a>0.0$.

(errno $3$)

On entry, $b=⟨value⟩$.

Constraint: $b>0.0$.

(errno $4$)

On entry, $statecomm$[‘state’] vector has been corrupted or not initialized.

Warns

NagAlgorithmicWarning

(errno $7$)

On entry, $a=⟨value⟩$.

For very small shape parameter values, variates are approximate.

Notes

The gamma distribution has PDF (probability density function)

$$\begin{array}{c}\begin{array}{cc}f\left(x\right)=\frac{1}{b^a\Gamma \left(a\right)}{x}^{a-1}{e}^{-x/b}& \text{if}x\ge 0\text{;}a,b>0\\ & \\ f\left(x\right)=0& \text{otherwise.}\end{array}\end{array}$$

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

1. 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 $f_2\left(x\right)=(1-c)e^{-(x-t)}$, 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 $t=1$;

2. 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;

3. 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 init_repeat() (for a repeatable sequence if computed sequentially) or init_nonrepeat() (for a non-repeatable sequence) must be called prior to the first call to dist_gamma.

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