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NAG Library Routine Document

g01eff (prob_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

1

Purpose

g01eff returns the lower or upper tail probability of the gamma distribution, with parameters

α

and

β

, via the routine name.

2

Specification

Fortran Interface

Function g01eff (

tail, g, a, b, ifail)

Real (Kind=nag_wp)	::	g01eff
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	g, a, b
Character (1), Intent (In)	::	tail

C Header Interface

#include nagmk26.h

double

g01eff_ ( const char *tail, const double *g, const double *a, const double *b, Integer *ifail, const Charlen length_tail)

3

Description

The lower tail probability for the gamma distribution with parameters

α

and

β

P (G \leq g)

, is defined by:

P (G \leq g; α, β) = \frac{1}{β^{α} Γ (α)} \int_{0}^{g} G^{α - 1} e^{- G / β} d G, α > 0.0, ​ β > 0.0 .

The mean of the distribution is

α β

and its variance is

α β^{2}

. The transformation

Z = \frac{G}{β}

is applied to yield the following incomplete gamma function in normalized form,

P (G \leq g; α, β) = P (Z \leq g / β : α, 1.0) = \frac{1}{Γ (α)} \int_{0}^{g / β} Z^{α - 1} e^{- Z} d Z .

This is then evaluated using s14baf.

4

References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

5

Arguments

1: $tail$ – Character(1)Input

On entry: indicates whether an upper or lower tail probability is required.

$tail ='L'$: The lower tail probability is returned, that is $P (G \leq g : α, β)$ .
$tail ='U'$: The upper tail probability is returned, that is $P (G \geq g : α, β)$ .

Constraint:

tail ='L'

'U'

2: $g$ – Real (Kind=nag_wp)Input

On entry:

g

, the value of the gamma variate.

Constraint:

g \geq 0.0

3: $a$ – Real (Kind=nag_wp)Input

On entry: the parameter

α

of the gamma distribution.

Constraint:

a > 0.0

4: $b$ – Real (Kind=nag_wp)Input

On entry: the parameter

β

of the gamma distribution.

Constraint:

b > 0.0

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, if you are not familiar with this argument, the recommended value is

0

. 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).

Errors or warnings detected by the routine:

ifail = 1

2

3

4

on exit, then g01eff returns

0.0

$ifail = 1$: On entry, $tail = 〈value〉$ .
Constraint: $tail ='L'$ or $'U'$ .

$ifail = 2$: On entry, $g = 〈value〉$ .
Constraint: $g \geq 0.0$ .

$ifail = 3$: On entry, $a = 〈value〉$ and $b = 〈value〉$ .
Constraint: $a > 0.0$ and $b > 0.0$ .

$ifail = 4$: The algorithm has failed to converge in $〈value〉$ iterations. The probability returned should be a reasonable approximation to the solution.

$ifail = - 99$: 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

The result should have a relative accuracy of machine precision. There are rare occasions when the relative accuracy attained is somewhat less than machine precision but the error should not exceed more than

1

2

decimal places. Note also that there is a limit of

18

decimal places on the achievable accuracy, because constants in s14baf are given to this precision.

8

Parallelism and Performance

g01eff is not threaded in any implementation.

9

Further Comments

The time taken by g01eff varies slightly with the input arguments g, a and b.

10

Example

This example reads in values from a number of gamma distributions and computes the associated lower tail probabilities.

NAG Library Routine Document

g01eff (prob_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

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