naginterfaces.library.rand.dist_​beta

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

dist_beta generates a vector of pseudorandom numbers taken from a beta distribution with parameters $a$ and $b$.

For full information please refer to the NAG Library document for g05sb

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g05/g05sbf.html

Parameters

nint

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

afloat

$a$, the parameter of the beta distribution.

bfloat

$b$, the parameter of the beta 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 beta 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.

Notes

The beta distribution has PDF (probability density function)

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

One of four algorithms is used to generate the variates depending on the values of $a$ and $b$. Let $\alpha$ be the maximum and $\beta$ be the minimum of $a$ and $b$. Then the algorithms are as follows:

1. if $\alpha <0.5$, Johnk’s algorithm is used, see for example Dagpunar (1988). This generates the beta variate as $u_1^{1/a}/\left(\begin{array}{c}{u}_1^{1/a}+u_2^{1/b}\end{array}\right)$, where $u_1$ and $u_2$ are uniformly distributed random variates;

2. if $\beta >1$, the algorithm BB given by Cheng (1978) is used. This involves the generation of an observation from a beta distribution of the second kind by the envelope rejection method using a log-logistic target distribution and then transforming it to a beta variate;

3. if $\alpha >1$ and $\beta <1$, the switching algorithm given by Atkinson (1979) is used. The two target distributions used are $f_1\left(x\right)=\beta x^{\beta }$ and $f_2\left(x\right)=\alpha {(1-x)}^{\beta -1}$, along with the approximation to the switching parameter of $t=(1-\beta )/(\alpha +1-\beta )$;

4. in all other cases, Cheng’s BC algorithm (see Cheng (1978)) is used with modifications suggested by Dagpunar (1988). This algorithm is similar to BB, used when $\beta >1$, but is tuned for small values of $a$ and $b$.

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_beta.

References

Atkinson, A C, 1979, A family of switching algorithms for the computer generation of beta random variates, Biometrika (66), 141–5

Cheng, R C H, 1978, Generating beta variates with nonintegral shape parameters, Comm. ACM (21), 317–322

Dagpunar, J, 1988, Principles of Random Variate Generation, Oxford University Press

Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth