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 <nag.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)

The routine may be called by the names g01sff or nagf_stat_prob_gamma_vector.

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: $ltail$ – Integer Input

On entry: the length of the array tail.

Constraint:

ltail > 0

2: $tail (ltail)$ – Character(1) array Input

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: $\lg$ – Integer Input

On entry: the length of the array g.

Constraint:

\lg > 0

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

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: $la$ – Integer Input

On entry: the length of the array a.

Constraint:

la > 0

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

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: $lb$ – Integer Input

On entry: the length of the array b.

Constraint:

lb > 0

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

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) array Output

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 array Output

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: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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.

g01sf: FL CL CPP AD

NAG FL Interfaceg01sff (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

NAG FL Interface
g01sff (prob_gamma_vector)