# NAG Library Function Document

## 1Purpose

nag_deviates_gamma_dist (g01ffc) returns the deviate associated with the given lower tail probability of the gamma distribution.

## 2Specification

 #include #include
 double nag_deviates_gamma_dist (double p, double a, double b, double tol, NagError *fail)

## 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:    $\mathbf{p}$doubleInput
On entry: $p$, the lower tail probability from the required gamma distribution.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
2:    $\mathbf{a}$doubleInput
On entry: $\alpha$, the shape parameter of the gamma distribution.
Constraint: $0.0<{\mathbf{a}}\le {10}^{6}$.
3:    $\mathbf{b}$doubleInput
On entry: $\beta$, the scale parameter of the gamma distribution.
Constraint: ${\mathbf{b}}>0.0$.
4:    $\mathbf{tol}$doubleInput
On entry: the relative accuracy required by you in the results. The smallest recommended value is $50×\delta$, where . If nag_deviates_gamma_dist (g01ffc) is entered with tol less than $50×\delta$ or greater or equal to $1.0$, then $50×\delta$ is used instead.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

On any of the error conditions listed below, except ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_ALG_NOT_CONV, nag_deviates_gamma_dist (g01ffc) returns $0.0$.
NE_ALG_NOT_CONV
The algorithm has failed to converge in 100 iterations. A larger value of tol should be tried. The result may be a reasonable approximation.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_GAM_NOT_CONV
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_PROBAB_CLOSE_TO_TAIL
The probability is too close to $0.0$ for the given a to enable the result to be calculated.
NE_REAL_ARG_GE
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}<1.0$.
NE_REAL_ARG_GT
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}\le {10}^{6}$.
NE_REAL_ARG_LE
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}>0.0$.
NE_REAL_ARG_LT
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}\ge 0.0$.

## 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

nag_deviates_gamma_dist (g01ffc) 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 (g01ffce.c)

### 10.2Program Data

Program Data (g01ffce.d)

### 10.3Program Results

Program Results (g01ffce.r)

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