NAG Library Function Document

nag_random_beta (g05fec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_random_beta (g05fec) generates a vector of pseudorandom variates from a beta distribution with arguments

a

and

b

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_random_beta (double a, double b, Integer n, double x[], NagError *fail)

3 Description

The beta distribution has PDF (probability density function):

\begin{array}{l} f (x) = \frac{Γ (a + b)}{Γ (a) Γ (b)} x^{a - 1} {(1 - x)}^{b - 1} & if ​ 0 \leq x \leq 1; a, b > 0.0 \\ f (x) = 0 & otherwise. \end{array}

One of four algorithms is used to generate the variates depending on the values of

a

and

b

. Let

α

be the maximum and

β

be the minimum of

a

and

b

. Then the algorithms are as follows:

α < 0.5

Jöhnk's algorithm is used, see for example Dagpunar (1988). This generates the beta variate as

u_{1}^{1 / a} / (u_{1}^{1 / a} + u_{2}^{1 / b})

, where

u_{1}

and

u_{2}

are uniformly distributed random variates.

β > 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.

α > 1

and

β < 1

The switching algorithm given by Atkinson (1979) is used. The two target distributions used are

f_{1} (x) = β x^{β}

and

f_{2} (x) = α {(1 - x)}^{β - 1}

, along with the approximation to the switching argument of

t = (1 - β) / (α + 1 - β)

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

β > 1

, but is tuned for small values of

a

and

b

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: a – doubleInput: On entry: the argument, $a$ , of the beta distribution.
Constraint: $a > 0.0$ .
2: b – doubleInput: On entry: the argument, $b$ , of the beta 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 beta 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,

y

, from the beta distribution of the second kind from an observation,

x

, generated by nag_random_beta (g05fec) the transformation,

y = x / (1 - x)

, may be used.

To generate an observation,

y

, from an

F

-distribution with degrees of freedom

v_{1}

and

v_{2}

generate an observation from the beta distribution with arguments

v_{1} / 2

and

v_{2} / 2

and use the transformation

y = v_{2} x / v_{1} (1 - x)

9 Example

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

a = 2.0

and

b = 2.0

, generated by nag_random_beta (g05fec) after initialization by nag_random_init_repeatable (g05cbc).

9.1 Program Text

Program Text (g05fece.c)

9.2 Program Data

None.

9.3 Program Results

Program Results (g05fece.r)

nag_random_beta (g05fec) (PDF version)

g05 Chapter Contents

g05 Chapter Introduction

NAG C Library Manual

NAG Library Function Documentnag_random_beta (g05fec)