NAG Library Routine Document

G01SFF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− 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

G01SFF returns a number of lower or upper tail probabilities for the gamma distribution.

2 Specification

SUBROUTINE G01SFF ( LTAIL, TAIL, LG, G, LA, A, LB, B, P, IVALID, IFAIL)

INTEGER LTAIL, LG, LA, LB, IVALID(*), IFAIL

REAL (KIND=nag_wp) G(LG), A(LA), B(LB), P(*)

CHARACTER(1) TAIL(LTAIL)

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 parameters 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 Parameters

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 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, if you are not familiar with this parameter, 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: $LB > 0$ .

$IFAIL = - 999$: Dynamic memory allocation failed.

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 Further Comments

The time taken by G01SFF to calculate each probability varies slightly with the input parameters

g_{i}

α_{i}

and

β_{i}

9 Example

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

NAG Library Routine DocumentG01SFF

+− 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

NAG Library Routine Document

G01SFF