NAG Library Routine Document

G05SGF

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

G05SGF generates a vector of pseudorandom numbers from an exponential mix distribution composed of

m

exponential distributions each having a mean

a_{i}

and weight

w_{i}

2 Specification

SUBROUTINE G05SGF ( N, NMIX, A, WGT, STATE, X, IFAIL)

INTEGER N, NMIX, STATE(*), IFAIL

REAL (KIND=nag_wp) A(NMIX), WGT(NMIX), X(N)

3 Description

The distribution has PDF (probability density function)

\begin{matrix} f (x) = \sum_{i = 1}^{m} \frac{1}{a_{i}} w_{i} e^{- x / a_{i}} & if ​ x \geq 0, \\ f (x) = 0 & otherwise, \end{matrix}

where

\sum_{i = 1}^{m} w_{i} = 1

and

a_{i} > 0

w_{i} \geq 0

G05SGF returns the values

x_{i}

by selecting, with probability

w_{j}

, random variates from an exponential distribution with parameter

a_{j}

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 G05SGF.

4 References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

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: NMIX – INTEGERInput

On entry:

m

, the number of exponential distributions in the mix.

Constraint:

NMIX \geq 1

3: A(NMIX) – REAL (KIND=nag_wp) arrayInput

On entry: the

m

parameters

a_{i}

for the

m

exponential distributions in the mix.

Constraint:

A (i) > 0.0

, for

i = 1, 2, \dots, NMIX

4: WGT(NMIX) – REAL (KIND=nag_wp) arrayInput

On entry: the

m

weights

w_{i}

for the

m

exponential distributions in the mix.

Constraints:

$\sum_{i = 1}^{m} WGT (i) = 1.0$ ;
$WGT (i) \geq 0.0$ , for $i = 1, 2, \dots, m$ .

5: 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.

6: X(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the

n

pseudorandom numbers from the specified exponential mix distribution.

7: 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, $NMIX \leq 0$ .

$IFAIL = 3$: On entry, $A (i) \leq 0.0$ for at least one $A (i)$ .

$IFAIL = 4$: On entry, $WGT (i) < 0.0$ for at least one $WGT (i)$ .

On entry, $\sum_{i = 1}^{NMIX} WGT (i) \neq 1.0$ .

$IFAIL = 5$: On entry, STATE vector was not initialized or has been corrupted.

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

This example prints the first five pseudorandom numbers from an exponential mix distribution comprising three exponential distributions with parameters

a_{1} = 1.0

a_{2} = 5.0

and

a_{3} = 2.0

, and with respective weights

0.5

0.3

and

0.2

. The numbers are generated by a single call to G05SGF, after initialization by G05KFF.

NAG Library Routine DocumentG05SGF

+− 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

NAG Library Routine Document

G05SGF