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NAG Toolbox: nag_rand_dist_beta (g05sb)

Purpose

nag_rand_dist_beta (g05sb) generates a vector of pseudorandom numbers taken from a beta distribution with parameters aa and bb.

Syntax

[state, x, ifail] = g05sb(n, a, b, state)
[state, x, ifail] = nag_rand_dist_beta(n, a, b, state)

Description

The beta distribution has PDF (probability density function)
f(x) = ( Γ(a + b) )/( Γ(a) Γ(b) ) xa1 (1x)b1 if  0x1 ; ​ a,b > 0 ,
f(x) = 0 otherwise.
f(x) = Γ(a+b) Γ(a) Γ(b) xa-1 (1-x) b-1 if  0x1 ; ​ a,b>0 , f(x)=0 otherwise.
One of four algorithms is used to generate the variates depending on the values of aa and bb. Let αα be the maximum and ββ be the minimum of aa and bb. Then the algorithms are as follows:
(i) if α < 0.5α<0.5, Johnk's algorithm is used, see for example Dagpunar (1988). This generates the beta variate as u11 / a /
(u11 / a + u21 / b)
u11/a/ u11/a+u21/b , where u1u1 and u2u2 are uniformly distributed random variates;
(ii) if β > 1β>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 > 1α>1 and β < 1β<1, the switching algorithm given by Atkinson (1979) is used. The two target distributions used are f1(x) = βxβf1(x)=βxβ and f2(x) = α(1x)β1f2(x)=α(1-x)β-1, along with the approximation to the switching parameter of t = (1β) / (α + 1β)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 > 1β>1, but is tuned for small values of aa and bb.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_dist_beta (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

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
nn, the number of pseudorandom numbers to be generated.
Constraint: n0n0.
2:     a – double scalar
aa, the parameter of the beta distribution.
Constraint: a > 0.0a>0.0.
3:     b – double scalar
bb, the parameter of the beta distribution.
Constraint: b > 0.0b>0.0.
4:     state( : :) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     state( : :) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains updated information on the state of the generator.
2:     x(n) – double array
The nn pseudorandom numbers from the specified beta distribution.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry, n < 0n<0.
  ifail = 2ifail=2
On entry, a0.0a0.0.
  ifail = 3ifail=3
On entry, b0.0b0.0.
  ifail = 4ifail=4
On entry,state vector was not initialized or has been corrupted.

Accuracy

Not applicable.

Further Comments

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

Example

function nag_rand_dist_beta_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
a = 2;
b = 2;
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_beta(n, a, b, state)
 

state =

                   17
                 1234
                    1
                    0
                 9910
                16740
                20386
                10757
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    0.5977
    0.6818
    0.1797
    0.4174
    0.4987


ifail =

                    0


function g05sb_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
a = 2;
b = 2;
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sb(n, a, b, state)
 

state =

                   17
                 1234
                    1
                    0
                 9910
                16740
                20386
                10757
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    0.5977
    0.6818
    0.1797
    0.4174
    0.4987


ifail =

                    0



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Chapter Introduction
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