NAG Library Routine Document

G01FFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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

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 Parameters

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 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 $- 1 or 1$ 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 $IFAIL \neq 0$ 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,	$P < 0.0$ ,
or	$P \geq 1.0$ ,

$IFAIL = 2$

On entry,	$A \leq 0.0$ ,
or	$A > 10^{6}$ ,
or	$B \leq 0.0$

$IFAIL = 3$: P is too close to $0.0$ or $1.0$ to enable the result to be calculated.

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

$IFAIL = 5$: 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

α

or very small values of

p

there may be some loss of accuracy.

8 Further Comments

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.

NAG Library Routine DocumentG01FFF