nag_rngs_beta (g05lec) (PDF version)
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g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rngs_beta (g05lec)

## 1  Purpose

nag_rngs_beta (g05lec) generates a vector of pseudorandom numbers taken from a beta distribution with $a$ and $b$.

## 2  Specification

 #include #include
 void nag_rngs_beta (double a, double b, Integer n, double x[], Integer igen, Integer iseed[], NagError *fail)

## 3  Description

The beta distribution has PDF (probability density function)
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:
 (i) If $\alpha <0.5$, Jhnk'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; (ii) 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; (iii) 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 ${f}_{2}\left(x\right)=\alpha {\left(1-x\right)}^{\beta -1}$, along with the approximation to the switching parameter of $t=\left(1-\beta \right)/\left(\alpha +1-\beta \right)$; (iv) 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 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_beta (g05lec).

## 4  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

## 5  Arguments

1:     adoubleInput
On entry: $a$, the parameter of the beta distribution.
Constraint: ${\mathbf{a}}>0.0$.
2:     bdoubleInput
On entry: $b$, the parameter of the beta distribution.
Constraint: ${\mathbf{b}}>0.0$.
3:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
4:     x[n]doubleOutput
On exit: the $n$ pseudorandom numbers from the specified beta distribution.
5:     igenIntegerInput
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).
6:     iseed[$4$]IntegerCommunication Array
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.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_BAD_PARAM
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_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.

## 8  Further Comments

To generate an observation, $y$, from the beta distribution of the second kind from an observation, $x$, generated by nag_rngs_beta (g05lec) the transformation, $y=x/\left(1-x\right)$, may be used.

## 9  Example

This example prints a set of five pseudorandom numbers from a beta distribution with parameters $a=2.0$ and $b=2.0$, generated by a single call to nag_rngs_beta (g05lec), after initialization by nag_rngs_init_repeatable (g05kbc).

### 9.1  Program Text

Program Text (g05lece.c)

None.

### 9.3  Program Results

Program Results (g05lece.r)

nag_rngs_beta (g05lec) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual