NAG Library Function Document

nag_rngs_students_t (g05lbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_rngs_students_t (g05lbc) 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_rngs_students_t (Integer df, Integer n, double x[], Integer igen, Integer iseed[], 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_rngs_students_t (g05lbc) calculates the values

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

where the

y_{i}

are generated by nag_rngs_normal (g05lac) from a Normal distribution with mean

0

and variance

1.0

, and the

z_{i}

are generated by nag_rngs_gamma (g05lfc) from a gamma distribution 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_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_students_t (g05lbc).

4 References

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

5 Arguments

1: df – IntegerInput: On entry: $ν$ , the number of degrees of freedom of the distribution.
Constraint: $df \geq 1$ .
2: n – IntegerInput: On entry: $n$ , the number of pseudorandom numbers to be generated.
Constraint: $n \geq 0$ .
3: x[n] – doubleOutput: On exit: the $n$ pseudorandom numbers from the specified Student's $t$ -distribution.
4: igen – IntegerInput: On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).
5: iseed[ $4$ ] – IntegerCommunication Array: On entry: contains values which define the current state of the selected generator.
On exit: contains updated values defining the new state of the selected generator.
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, $df = 〈value〉$ .
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.

7 Accuracy

Not applicable.

8 Further Comments

The time taken by nag_rngs_students_t (g05lbc) increases with

ν

9 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_rngs_students_t (g05lbc), after initialization by nag_rngs_init_repeatable (g05kbc).