NAG Library Routine Document

g01fff (inv_cdf_gamma)

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NAG Library Manual, Mark 26

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G01 (stat) Chapter Contents

G01 (stat) Chapter Introduction

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

g01fff returns the deviate associated with the given lower tail probability of the gamma distribution.

2

Specification

Fortran Interface

Function g01fff (

p, a, b, tol, ifail)

Real (Kind=nag_wp)	::	g01fff
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	p, a, b, tol

C Header Interface

#include <nagmk26.h>

double

g01fff_ (const double *p, const double *a, const double *b, const double *tol, Integer *ifail)

3

Description

The deviate,

g_{p}

, associated with the lower tail probability,

p

, of the gamma distribution with shape parameter

α

and scale parameter

β

, is defined as the solution to

P (G \leq g_{p} : α, β) = p = \frac{1}{β^{α} Γ (α)} \int_{0}^{g_{p}} e^{- G / β} G^{α - 1} d G, 0 \leq g_{p} < \infty; α, β > 0 .

The method used is described by Best and Roberts (1975) making use of the relationship between the gamma distribution and the

χ^{2}

-distribution.

Let

y = 2 \frac{g_{p}}{β}

. The required

y

is found from the Taylor series expansion

y = y_{0} + \sum_{r} \frac{C_{r} (y_{0})}{r!} {(\frac{E}{ϕ (y_{0})})}^{r},

where

y_{0}

is a starting approximation

$C_{1} (u) = 1$ ,
$C_{r + 1} (u) = (r Ψ + \frac{d}{d u}) C_{r} (u)$ ,
$Ψ = \frac{1}{2} - \frac{α - 1}{u}$ ,
$E = p - \int_{0}^{y_{0}} ϕ (u) d u$ ,
$ϕ (u) = \frac{1}{2^{α} Γ (α)} e^{- u / 2} u^{α - 1}$ .

For most values of

p

and

α

the starting value

y_{01} = 2 α {(z \sqrt{\frac{1}{9 α}} + 1 - \frac{1}{9 α})}^{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,

y_{02} = {(p α 2^{α} Γ (α))}^{1 / α}

is used.

For large

p

values, when

y_{01} > 4.4 α + 6.0

y_{03} = - 2 [\ln (1 - p) - (α - 1) \ln (\frac{1}{2} y_{01}) + \ln (Γ (α))]

is found to be a better starting value than

y_{01}

For small

α

(α \leq 0.16)

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

χ^{2}

distribution Appl. Statist. 24 385–388

5

Arguments

1: $p$ – Real (Kind=nag_wp)Input: On entry: $p$ , the lower tail probability from the required gamma distribution.

Constraint: $0.0 \leq p < 1.0$ .
2: $a$ – Real (Kind=nag_wp)Input: On entry: $α$ , the shape parameter of the gamma distribution.

Constraint: $0.0 < a \leq 10^{6}$ .
3: $b$ – Real (Kind=nag_wp)Input: On entry: $β$ , the scale parameter of the gamma distribution.

Constraint: $b > 0.0$ .
4: $tol$ – Real (Kind=nag_wp)Input: On entry: the relative accuracy required by you in the results. The smallest recommended value is $50 \times δ$ , where $δ = \max (10^{- 18}, machine precision)$ . If g01fff is entered with tol less than $50 \times δ$ or greater or equal to $1.0$ , then $50 \times δ$ is used instead.
5: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . 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 $- 1 or 1$ 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 $ifail \neq 0$ on exit, the recommended value is $- 1$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 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

ifail = 1

2

3

5

, then g01fff returns

0.0

$ifail = 1$: On entry, $p = 〈value〉$ .
Constraint: $p < 1.0$ .

On entry, $p = 〈value〉$ .
Constraint: $p \geq 0.0$ .

$ifail = 2$: On entry, $a = 〈value〉$ .
Constraint: $a > 0.0$ .

On entry, $a = 〈value〉$ .
Constraint: $a \leq 10^{6}$ .

On entry, $b = 〈value〉$ .
Constraint: $b > 0.0$ .

$ifail = 3$: The probability is too close to $0.0$ for the given a to enable the result to be calculated.

$ifail = 4$: The algorithm has failed to converge in $100$ iterations. A larger value of tol should be tried. The result may be a reasonable approximation.

$ifail = 5$: The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

In most cases the relative accuracy of the results should be as specified by tol. However, for very small values of

α

or very small values of

p

there may be some loss of accuracy.

8

Parallelism and Performance

g01fff is not threaded in any implementation.

9

Further Comments

None.

10

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.

NAG Library Routine Document

g01fff (inv_cdf_gamma)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results