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NAG Toolbox

NAG Toolbox: nag_rand_dist_students_t (g05sn)

Purpose

nag_rand_dist_students_t (g05sn) generates a vector of pseudorandom numbers taken from a Student's tt-distribution with νν degrees of freedom.

Syntax

[state, x, ifail] = g05sn(n, df, state)
[state, x, ifail] = nag_rand_dist_students_t(n, df, state)

Description

The distribution has PDF (probability density function)
f(x) = ( ((ν1)/2) ! )/(((1/2)ν1) ! sqrt(πν) (1 + (x2)/ν)(1/2)(ν + 1)).
f(x)= (ν-12) ! (12ν-1)!πν (1+x2ν) 12(ν+1) .
nag_rand_dist_students_t (g05sn) calculates the values
yisqrt(ν/(zi)),   i = 1,,n
yiνzi,   i= 1,,n
where the yiyi are generated by nag_rand_dist_normal (g05sk) from a Normal distribution with mean 00 and variance 1.01.0, and the zizi are generated by nag_rand_dist_gamma (g05sj) from a gamma distribution with parameters (1/2)ν12ν and 22 (i.e., from a χ2χ2-distribution with νν degrees of freedom).
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_students_t (g05sn).

References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
nn, the number of pseudorandom numbers to be generated.
Constraint: n0n0.
2:     df – int64int32nag_int scalar
νν, the number of degrees of freedom of the distribution.
Constraint: df1 df1 .
3:     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 Student's tt-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, df < 1df<1.
  ifail = 3ifail=3
On entry,state vector was not initialized or has been corrupted.

Accuracy

Not applicable.

Further Comments

The time taken by nag_rand_dist_students_t (g05sn) increases with νν.

Example

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

state =

                   17
                 1234
                    1
                    0
                 6694
                27818
                10435
                15383
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    0.3849
   -0.9461
   -2.2814
    0.1127
    0.5272


ifail =

                    0


function g05sn_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
df = int64(5);
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sn(n, df, state)
 

state =

                   17
                 1234
                    1
                    0
                 6694
                27818
                10435
                15383
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    0.3849
   -0.9461
   -2.2814
    0.1127
    0.5272


ifail =

                    0



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