# NAG Library Routine Document

## 1Purpose

g01fff returns the deviate associated with the given lower tail probability of the gamma distribution, via the routine name.

## 2Specification

Fortran Interface
 Function g01fff ( p, a, b, tol,
 Real (Kind=nag_wp) :: g01fff Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: p, a, b, tol
#include nagmk26.h
 double g01fff_ (const double *p, const double *a, const double *b, const double *tol, Integer *ifail)

## 3Description

The deviate, ${g}_{p}$, associated with the lower tail probability, $p$, of the gamma distribution with shape parameter $\alpha$ and scale parameter $\beta$, is defined as the solution to
 $PG≤gp:α,β=p=1βαΓα ∫0gpe-G/βGα-1dG, 0≤gp<∞;α,β>0.$
The method used is described by Best and Roberts (1975) making use of the relationship between the gamma distribution and the ${\chi }^{2}$-distribution.
Let $y=2\frac{{g}_{p}}{\beta }$. The required $y$ is found from the Taylor series expansion
 $y=y0+∑rCry0 r! Eϕy0 r,$
where ${y}_{0}$ is a starting approximation
• ${C}_{1}\left(u\right)=1$,
• ${C}_{r+1}\left(u\right)=\left(r\Psi +\frac{d}{du}\right){C}_{r}\left(u\right)$,
• $\Psi =\frac{1}{2}-\frac{\alpha -1}{u}$,
• $E=p-\underset{0}{\overset{{y}_{0}}{\int }}\varphi \left(u\right)du$,
• $\varphi \left(u\right)=\frac{1}{{2}^{\alpha }\Gamma \left(\alpha \right)}{e}^{-u/2}{u}^{\alpha -1}$.
For most values of $p$ and $\alpha$ the starting value
 $y01=2α z⁢19α +1-19α 3$
is used, where $z$ is the deviate associated with a lower tail probability of $p$ for the standard Normal distribution.
For $p$ close to zero,
 $y02= pα2αΓ α 1/α$
is used.
For large $p$ values, when ${y}_{01}>4.4\alpha +6.0$,
 $y03=-2ln1-p-α-1ln12y01+lnΓ α$
is found to be a better starting value than ${y}_{01}$.
For small $\alpha$ $\left(\alpha \le 0.16\right)$, $p$ is expressed in terms of an approximation to the exponential integral and ${y}_{04}$ is found by Newton–Raphson iterations.
Seven terms of the Taylor series are used to refine the starting approximation, repeating the process if necessary until the required accuracy is obtained.

## 4References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the ${\chi }^{2}$ distribution Appl. Statist. 24 385–388

## 5Arguments

1:     $\mathbf{p}$ – Real (Kind=nag_wp)Input
On entry: $p$, the lower tail probability from the required gamma distribution.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
2:     $\mathbf{a}$ – Real (Kind=nag_wp)Input
On entry: $\alpha$, the shape parameter of the gamma distribution.
Constraint: $0.0<{\mathbf{a}}\le {10}^{6}$.
3:     $\mathbf{b}$ – Real (Kind=nag_wp)Input
On entry: $\beta$, the scale parameter of the gamma distribution.
Constraint: ${\mathbf{b}}>0.0$.
4:     $\mathbf{tol}$ – Real (Kind=nag_wp)Input
On entry: the relative accuracy required by you in the results. The smallest recommended value is $50×\delta$, where . If g01fff is entered with tol less than $50×\delta$ or greater or equal to $1.0$, then $50×\delta$ is used instead.
5:     $\mathbf{ifail}$ – 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, because for this routine the values of the output arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ 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).
Note: g01fff may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
If on exit ${\mathbf{ifail}}={\mathbf{1}}$, ${\mathbf{2}}$, ${\mathbf{3}}$ or ${\mathbf{5}}$, then g01fff returns $0.0$.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}\ge 0.0$.
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}<1.0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}\le {10}^{6}$.
On entry, ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}>0.0$.
${\mathbf{ifail}}=3$
The probability is too close to $0.0$ for the given a to enable the result to be calculated.
${\mathbf{ifail}}=4$
The algorithm has failed to converge in 100 iterations. A larger value of tol should be tried. The result may be a reasonable approximation.
${\mathbf{ifail}}=5$
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
${\mathbf{ifail}}=-99$
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

In most cases the relative accuracy of the results should be as specified by tol. However, for very small values of $\alpha$ or very small values of $p$ there may be some loss of accuracy.

## 8Parallelism and Performance

g01fff is not threaded in any implementation.

None.

## 10Example

This example reads lower tail probabilities for several gamma distributions, and calculates and prints the corresponding deviates until the end of data is reached.

### 10.1Program Text

Program Text (g01fffe.f90)

### 10.2Program Data

Program Data (g01fffe.d)

### 10.3Program Results

Program Results (g01fffe.r)

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