g05sb generates a vector of pseudorandom numbers taken from a beta distribution with parameters a and b.

Syntax

C#
public static void g05sb(
	int n,
	double a,
	double b,
	G05..::..G05State g05state,
	double[] x,
	out int ifail
)
Visual Basic
Public Shared Sub g05sb ( _
	n As Integer, _
	a As Double, _
	b As Double, _
	g05state As G05..::..G05State, _
	x As Double(), _
	<OutAttribute> ByRef ifail As Integer _
)
Visual C++
public:
static void g05sb(
	int n, 
	double a, 
	double b, 
	G05..::..G05State^ g05state, 
	array<double>^ x, 
	[OutAttribute] int% ifail
)
F#
static member g05sb : 
        n : int * 
        a : float * 
        b : float * 
        g05state : G05..::..G05State * 
        x : float[] * 
        ifail : int byref -> unit

Parameters

n
Type: System..::..Int32
On entry: n, the number of pseudorandom numbers to be generated.
Constraint: n≥0.
a
Type: System..::..Double
On entry: a, the parameter of the beta distribution.
Constraint: a>0.0.
b
Type: System..::..Double
On entry: b, the parameter of the beta distribution.
Constraint: b>0.0.
g05state
Type: NagLibrary..::..G05..::..G05State
An Object of type G05.G05State.
x
Type: array<System..::..Double>[]()[][]
An array of size [n]
On exit: the n pseudorandom numbers from the specified beta distribution.
ifail
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

The beta distribution has PDF (probability density function)
fx=Γa+bΓaΓbxa-11-xb-1if 0≤x≤1; ​a,b>0,fx=0otherwise.
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 u11/a/u11/a+u21/b, where u1 and u2 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 f1x=βxβ and f2x=α1-xβ-1, along with the approximation to the switching parameter 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 methods (G05KFF not in this release) (for a repeatable sequence if computed sequentially) or (G05KGF not in this release) (for a non-repeatable sequence) must be called prior to the first call to g05sb.

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

Error Indicators and Warnings

Errors or warnings detected by the method:
ifail=1
On entry, n<0.
ifail=2
On entry, a≤0.0.
ifail=3
On entry, b≤0.0.
ifail=4
On entry,state vector was not initialized or has been corrupted.
ifail=-9000
An error occured, see message report.
ifail=-8000
Negative dimension for array value
ifail=-6000
Invalid Parameters value

Accuracy

Not applicable.

Parallelism and Performance

None.

Further Comments

To generate an observation, y, from the beta distribution of the second kind from an observation, x, generated by g05sb the transformation, y=x/1-x, may be used.

Example

See Also