NAG Library Routine Document

Integer, Intent (In)	::	ltail, lg, la, lb
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(*)
Real (Kind=nag_wp), Intent (In)	::	g(lg), a(la), b(lb)
Real (Kind=nag_wp), Intent (Out)	::	p(*)
Character (1), Intent (In)	::	tail(ltail)

C Header Interface

#include <nagmk26.h>

void	g01sff_ (const Integer ltail, const char tail[], const Integer lg, const double g[], const Integer la, const double a[], const Integer lb, const double b[], double p[], Integer ivalid[], Integer *ifail, const Charlen length_tail)

3

Description

The lower tail probability for the gamma distribution with parameters

α_{i}

and

β_{i}

P (G_{i} \leq g_{i})

, is defined by:

P (G_{i} \leq g_{i} : α_{i}, β_{i}) = \frac{1}{{β_{i}}^{α_{i}} Γ (α_{i})} \int_{0}^{g_{i}} {G_{i}}^{α_{i} - 1} e^{- G_{i} / β_{i}} d G_{i}, α_{i} > 0.0, ​ β_{i} > 0.0 .

The mean of the distribution is

α_{i} β_{i}

and its variance is

α_{i} {β_{i}}^{2}

. The transformation

Z_{i} = \frac{G_{i}}{β_{i}}

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

P (G_{i} \leq g_{i} : α_{i}, β_{i}) = P (Z_{i} \leq g_{i} / β_{i} : α_{i}, 1.0) = \frac{1}{Γ (α_{i})} \int_{0}^{g_{i} / β_{i}} {Z_{i}}^{α_{i} - 1} e^{- Z_{i}} d Z_{i} .

This is then evaluated using s14baf.

The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

4

References

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

5

Arguments

1: $ltail$ – IntegerInput

On entry: the length of the array tail.

Constraint:

ltail > 0

2: $tail (ltail)$ – Character(1) arrayInput

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

j = ((i - 1) mod ltail) + 1

, for

i = 1, 2, \dots, \max (ltail, \lg, la, lb)

$tail (j) ='L'$: The lower tail probability is returned, i.e., $p_{i} = P (G_{i} \leq g_{i} : α_{i}, β_{i})$ .
$tail (j) ='U'$: The upper tail probability is returned, i.e., $p_{i} = P (G_{i} \geq g_{i} : α_{i}, β_{i})$ .

Constraint:

tail (j) ='L'

'U'

, for

j = 1, 2, \dots, ltail

3: $\lg$ – IntegerInput

On entry: the length of the array g.

Constraint:

\lg > 0

4: $g (\lg)$ – Real (Kind=nag_wp) arrayInput

On entry:

g_{i}

, the value of the gamma variate with

g_{i} = g (j)

j = ((i - 1) mod \lg) + 1

Constraint:

g (j) \geq 0.0

, for

j = 1, 2, \dots, \lg

5: $la$ – IntegerInput

On entry: the length of the array a.

Constraint:

la > 0

6: $a (la)$ – Real (Kind=nag_wp) arrayInput

On entry: the parameter

α_{i}

of the gamma distribution with

α_{i} = a (j)

j = ((i - 1) mod la) + 1

Constraint:

a (j) > 0.0

, for

j = 1, 2, \dots, la

7: $lb$ – IntegerInput

On entry: the length of the array b.

Constraint:

lb > 0

8: $b (lb)$ – Real (Kind=nag_wp) arrayInput

On entry: the parameter

β_{i}

of the gamma distribution with

β_{i} = b (j)

j = ((i - 1) mod lb) + 1

Constraint:

b (j) > 0.0

, for

j = 1, 2, \dots, lb

9: $p (*)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array p must be at least

\max (\lg, la, lb, ltail)

On exit:

p_{i}

, the probabilities of the beta distribution.

10: $ivalid (*)$ – Integer arrayOutput

Note: the dimension of the array ivalid must be at least

\max (\lg, la, lb, ltail)

On exit:

ivalid (i)

indicates any errors with the input arguments, with

$ivalid (i) = 0$

No error.

$ivalid (i) = 1$

On entry,

invalid value supplied in tail when calculating

p_{i}

$ivalid (i) = 2$

On entry,

g_{i} < 0.0

$ivalid (i) = 3$

On entry,	$α_{i} \leq 0.0$ ,
or	$β_{i} \leq 0.0$ .

$ivalid (i) = 4$

The solution did not converge in

600

iterations, see s14baf for details. The probability returned should be a reasonable approximation to the solution.

11: $ifail$ – 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:

$ifail = 1$: On entry, at least one value of g, a, b or tail was invalid, or the solution did not converge.
Check ivalid for more information.

$ifail = 2$: On entry, $array size = 〈value〉$ .
Constraint: $ltail > 0$ .

$ifail = 3$: On entry, $array size = 〈value〉$ .
Constraint: $\lg > 0$ .

$ifail = 4$: On entry, $array size = 〈value〉$ .
Constraint: $la > 0$ .

$ifail = 5$: On entry, $array size = 〈value〉$ .
Constraint: $lb > 0$ .

$ifail = - 99$: 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.

8

Parallelism and Performance

g01sff is not threaded in any implementation.

9

Further Comments

The time taken by g01sff to calculate each probability varies slightly with the input arguments

g_{i}

α_{i}

and

β_{i}

10

Example

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

NAG Library Routine Document

g01sff (prob_gamma_vector)

▸▿ 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