nag_rand_beta (g05sbc) : NAG Library, Mark 24

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, \\ 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:

(i)	if $α < 0.5$ , Johnk's algorithm is used, see for example Dagpunar (1988). This generates the beta variate as $u_{1}^{1 / a} / (\begin{matrix} u_{1}^{1 / a} + u_{2}^{1 / b} \end{matrix})$ , where $u_{1}$ and $u_{2}$ are uniformly distributed random variates;
(ii)	if $β > 1$ , 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;
(iii)	if $α > 1$ 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 $t = (1 - β) / (α + 1 - β)$ ;
(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 $β > 1$ , but is tuned for small values of $a$ and $b$ .

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_beta (g05sbc).

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: n – IntegerInput

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 0

2: a – doubleInput

On entry:

a

, the parameter of the beta distribution.

Constraint:

a > 0.0

3: b – doubleInput

On entry:

b

, the parameter of the beta distribution.

Constraint:

b > 0.0

4: state[

\dim

] – IntegerCommunication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to 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 beta distribution.

6: fail – NagError *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

⟨value⟩

had an illegal value.

NE_INT

On entry,

n = ⟨value⟩

.
Constraint:

n \geq 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,

a = ⟨value⟩

.
Constraint:

a > 0.0

On entry,

b = ⟨value⟩

.
Constraint:

b > 0.0

NAG Library Function Document

nag_rand_beta (g05sbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Documentnag_rand_beta (g05sbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_rand_beta (g05sbc)