NAG Library Routine Document

G05SHF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

G05SHF generates a vector of pseudorandom numbers taken from an

F

(or Fisher's variance ratio) distribution with

μ

and

ν

degrees of freedom.

2 Specification

SUBROUTINE G05SHF ( N, DF1, DF2, STATE, X, IFAIL)

INTEGER N, DF1, DF2, STATE(*), IFAIL

REAL (KIND=nag_wp) X(N)

3 Description

The distribution has PDF (probability density function)

\begin{matrix} f (x) = \frac{(\frac{μ + ν - 2}{2})! x^{\frac{1}{2} μ - 1}}{(\frac{1}{2} μ - 1)! (\frac{1}{2} ν - 1)! {(1 + \frac{μ}{ν} x)}^{\frac{1}{2} (μ + ν)}} \times {(\frac{μ}{ν})}^{\frac{1}{2} μ} & if ​ x > 0, \\ f (x) = 0 & otherwise. \end{matrix}

G05SHF calculates the values

\frac{ν y_{i}}{μ z_{i}}, i = 1, 2, \dots, n,

where

y_{i}

and

z_{i}

are generated by G05SJF from gamma distributions with parameters

(\frac{1}{2} μ, 2)

and

(\frac{1}{2} ν, 2)

respectively (i.e., from

χ^{2}

-distributions with

μ

and

ν

degrees of freedom).

One of the initialization routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05SHF.

4 References

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

5 Parameters

1: N – INTEGERInput: On entry: $n$ , the number of pseudorandom numbers to be generated.
Constraint: $N \geq 0$ .
2: DF1 – INTEGERInput: On entry: $μ$ , the number of degrees of freedom of the distribution.
Constraint: $DF1 \geq 1$ .
3: DF2 – INTEGERInput: On entry: $ν$ , the number of degrees of freedom of the distribution.
Constraint: $DF2 \geq 1$ .
4: STATE( $*$ ) – INTEGER arrayCommunication Array: Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.
On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.
5: X(N) – REAL (KIND=nag_wp) arrayOutput: On exit: the $n$ pseudorandom numbers from the specified $F$ -distribution.
6: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $N < 0$ .

$IFAIL = 2$: On entry, $DF1 < 1$ .

$IFAIL = 3$: On entry, $DF2 < 1$ .

$IFAIL = 4$: On entry, STATE vector was not initialized or has been corrupted.

7 Accuracy

Not applicable.

8 Further Comments

The time taken by G05SHF increases with

μ

and

ν

9 Example

This example prints five pseudorandom numbers from an

F

-distribution with two and three degrees of freedom, generated by a single call to G05SHF, after initialization by G05KFF.

NAG Library Routine DocumentG05SHF