# NAG Library Routine Document

## 1Purpose

g01sff returns a number of lower or upper tail probabilities for the gamma distribution.

## 2Specification

Fortran Interface
 Subroutine g01sff ( tail, lg, g, la, a, lb, b, p,
 Integer, Intent (In) :: ltail, lg, la, lb Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: g(lg), a(la), b(lb) Real (Kind=nag_wp), Intent (Out) :: p(*) Character (1), Intent (In) :: tail(ltail)
#include nagmk26.h
 void g01sff_ (const Integer *ltail, const char tail[], const Integer *lg, const double g[], const Integer *la, const double a[], const Integer *lb, const double b[], double p[], Integer ivalid[], Integer *ifail, const Charlen length_tail)

## 3Description

The lower tail probability for the gamma distribution with parameters ${\alpha }_{i}$ and ${\beta }_{i}$, $P\left({G}_{i}\le {g}_{i}\right)$, is defined by:
 $P Gi ≤ gi :αi,βi = 1 βi αi Γ αi ∫ 0 gi Gi αi-1 e -Gi/βi dGi , αi>0.0 , ​ βi>0.0 .$
The mean of the distribution is ${\alpha }_{i}{\beta }_{i}$ and its variance is ${\alpha }_{i}{{\beta }_{i}}^{2}$. The transformation ${Z}_{i}=\frac{{G}_{i}}{{\beta }_{i}}$ is applied to yield the following incomplete gamma function in normalized form,
 $P Gi ≤ gi :αi,βi = P Zi ≤ gi / βi :αi,1.0 = 1 Γ αi ∫ 0 gi / βi Zi αi-1 e -Zi dZi .$
This is then evaluated using s14baf.
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

## 4References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 5Arguments

1:     $\mathbf{ltail}$ – IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:     $\mathbf{tail}\left({\mathbf{ltail}}\right)$ – Character(1) arrayInput
On entry: indicates whether a lower or upper tail probability is required. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lg}},{\mathbf{la}},{\mathbf{lb}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({G}_{i}\le {g}_{i}:{\alpha }_{i},{\beta }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({G}_{i}\ge {g}_{i}:{\alpha }_{i},{\beta }_{i}\right)$.
Constraint: ${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$ or $\text{'U'}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:     $\mathbf{lg}$ – IntegerInput
On entry: the length of the array g.
Constraint: ${\mathbf{lg}}>0$.
4:     $\mathbf{g}\left({\mathbf{lg}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${g}_{i}$, the value of the gamma variate with ${g}_{i}={\mathbf{g}}\left(j\right)$, .
Constraint: ${\mathbf{g}}\left(\mathit{j}\right)\ge 0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lg}}$.
5:     $\mathbf{la}$ – IntegerInput
On entry: the length of the array a.
Constraint: ${\mathbf{la}}>0$.
6:     $\mathbf{a}\left({\mathbf{la}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the parameter ${\alpha }_{i}$ of the gamma distribution with ${\alpha }_{i}={\mathbf{a}}\left(j\right)$, .
Constraint: ${\mathbf{a}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
7:     $\mathbf{lb}$ – IntegerInput
On entry: the length of the array b.
Constraint: ${\mathbf{lb}}>0$.
8:     $\mathbf{b}\left({\mathbf{lb}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the parameter ${\beta }_{i}$ of the gamma distribution with ${\beta }_{i}={\mathbf{b}}\left(j\right)$, .
Constraint: ${\mathbf{b}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.
9:     $\mathbf{p}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array p must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lg}},{\mathbf{la}},{\mathbf{lb}},{\mathbf{ltail}}\right)$.
On exit: ${p}_{i}$, the probabilities of the beta distribution.
10:   $\mathbf{ivalid}\left(*\right)$ – Integer arrayOutput
Note: the dimension of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lg}},{\mathbf{la}},{\mathbf{lb}},{\mathbf{ltail}}\right)$.
On exit: ${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
 On entry, invalid value supplied in tail when calculating ${p}_{i}$.
${\mathbf{ivalid}}\left(i\right)=2$
 On entry, ${g}_{i}<0.0$.
${\mathbf{ivalid}}\left(i\right)=3$
 On entry, ${\alpha }_{i}\le 0.0$, or ${\beta }_{i}\le 0.0$.
${\mathbf{ivalid}}\left(i\right)=4$
The solution did not converge in $600$ iterations, see s14baf for details. The probability returned should be a reasonable approximation to the solution.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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, at least one value of g, a, b or tail was invalid, or the solution did not converge.
${\mathbf{ifail}}=2$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lg}}>0$.
${\mathbf{ifail}}=4$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{la}}>0$.
${\mathbf{ifail}}=5$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lb}}>0$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The result should have a relative accuracy of machine precision. There are rare occasions when the relative accuracy attained is somewhat less than machine precision but the error should not exceed more than $1$ or $2$ decimal places.

## 8Parallelism and Performance

g01sff is not threaded in any implementation.

The time taken by g01sff to calculate each probability varies slightly with the input arguments ${g}_{i}$, ${\alpha }_{i}$ and ${\beta }_{i}$.

## 10Example

This example reads in values from a number of gamma distributions and computes the associated lower tail probabilities.

### 10.1Program Text

Program Text (g01sffe.f90)

### 10.2Program Data

Program Data (g01sffe.d)

### 10.3Program Results

Program Results (g01sffe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017