G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05SGF

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.

## 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)
 $fx = ∑i=1m 1ai wi e-x/ai if ​x≥0, fx = 0 otherwise,$
where $\sum _{i=1}^{m}{w}_{i}=1$ and ${a}_{i}>0$, ${w}_{i}\ge 0$.
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: ${\mathbf{N}}\ge 0$.
2:     NMIX – INTEGERInput
On entry: $m$, the number of exponential distributions in the mix.
Constraint: ${\mathbf{NMIX}}\ge 1$.
3:     A(NMIX) – REAL (KIND=nag_wp) arrayInput
On entry: the $m$ parameters ${a}_{i}$ for the $m$ exponential distributions in the mix.
Constraint: ${\mathbf{A}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{NMIX}}$.
4:     WGT(NMIX) – REAL (KIND=nag_wp) arrayInput
On entry: the $m$ weights ${w}_{i}$ for the $m$ exponential distributions in the mix.
Constraints:
• $\sum _{\mathit{i}=1}^{m}{\mathbf{WGT}}\left(\mathit{i}\right)=1.0$;
• ${\mathbf{WGT}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,m$.
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 $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, ${\mathbf{N}}<0$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{NMIX}}\le 0$.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{A}}\left(i\right)\le 0.0$ for at least one ${\mathbf{A}}\left(i\right)$.
${\mathbf{IFAIL}}=4$
On entry, ${\mathbf{WGT}}\left(i\right)<0.0$ for at least one ${\mathbf{WGT}}\left(i\right)$.
On entry, $\sum _{i=1}^{{\mathbf{NMIX}}}{\mathbf{WGT}}\left(i\right)\ne 1.0$.
${\mathbf{IFAIL}}=5$
 On entry, STATE vector was not initialized or has been corrupted.

Not applicable.

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.

### 9.1  Program Text

Program Text (g05sgfe.f90)

### 9.2  Program Data

Program Data (g05sgfe.d)

### 9.3  Program Results

Program Results (g05sgfe.r)