G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01FFF

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

G01FFF returns the deviate associated with the given lower tail probability of the gamma distribution, via the routine name.

## 2  Specification

 FUNCTION G01FFF ( P, A, B, TOL, IFAIL)
 REAL (KIND=nag_wp) G01FFF
 INTEGER IFAIL REAL (KIND=nag_wp) P, A, B, TOL

## 3  Description

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.

## 4  References

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

## 5  Parameters

1:     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:     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:     B – REAL (KIND=nag_wp)Input
On entry: $\beta$, the scale parameter of the gamma distribution.
Constraint: ${\mathbf{B}}>0.0$.
4:     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:     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, because for this routine the values of the output parameters 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).

## 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).
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, ${\mathbf{P}}<0.0$, or ${\mathbf{P}}\ge 1.0$,
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{A}}\le 0.0$, or ${\mathbf{A}}>{10}^{6}$, or ${\mathbf{B}}\le 0.0$
${\mathbf{IFAIL}}=3$
P is too close to $0.0$ or $1.0$ to enable the result to be calculated.
${\mathbf{IFAIL}}=4$
The solution 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 to calculate the gamma function has failed to converge. This is an unlikely error exit.

## 7  Accuracy

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.

None.

## 9  Example

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

### 9.1  Program Text

Program Text (g01fffe.f90)

### 9.2  Program Data

Program Data (g01fffe.d)

### 9.3  Program Results

Program Results (g01fffe.r)