NAG Library Function Document

nag_rand_students_t (g05snc)

+− 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

1 Purpose

nag_rand_students_t (g05snc) generates a vector of pseudorandom numbers taken from a Student's

t

-distribution with

ν

degrees of freedom.

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rand_students_t (Integer n, Integer df, Integer state[], double x[], NagError *fail)

3 Description

The distribution has PDF (probability density function)

f (x) = \frac{(\frac{ν - 1}{2})!}{(\frac{1}{2} ν - 1)! \sqrt{π ν} {(1 + \frac{x^{2}}{ν})}^{\frac{1}{2} (ν + 1)}} .

nag_rand_students_t (g05snc) calculates the values

y_{i} \sqrt{\frac{ν}{z_{i}}}, i = 1, \dots, n

where the

y_{i}

are generated by nag_rand_normal (g05skc) with mean

0

and variance

1.0

, and the

z_{i}

are generated by nag_rand_gamma (g05sjc) with parameters

\frac{1}{2} ν

and

2

(i.e., from a

χ^{2}

-distribution with

ν

degrees of freedom).

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_students_t (g05snc).

4 References

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

5 Arguments

1: n – IntegerInput: On entry: $n$ , the number of pseudorandom numbers to be generated.
Constraint: $n \geq 0$ .
2: df – IntegerInput: On entry: $ν$ , the number of degrees of freedom of the distribution.
Constraint: $df \geq 1$ .
3: 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.
4: x[n] – doubleOutput: On exit: the $n$ pseudorandom numbers from the specified Student's $t$ -distribution.
5: 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, $df = ⟨value⟩$ .
Constraint: $df \geq 1$ .
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.

7 Accuracy

Not applicable.

8 Parallelism and Performance

nag_rand_students_t (g05snc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by nag_rand_students_t (g05snc) increases with

ν

10 Example

This example prints five pseudorandom numbers from a Student's

t

-distribution with five degrees of freedom, generated by a single call to nag_rand_students_t (g05snc), after initialization by nag_rand_init_repeatable (g05kfc).