NAG Library Routine Document

G01SEF

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

G01SEF computes a number of lower or upper tail probabilities for the beta distribution.

2 Specification

SUBROUTINE G01SEF ( LTAIL, TAIL, LBETA, BETA, LA, A, LB, B, P, IVALID, IFAIL)

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

REAL (KIND=nag_wp) BETA(LBETA), A(LA), B(LB), P(*)

CHARACTER(1) TAIL(LTAIL)

3 Description

The lower tail probability,

P (B_{i} \leq β_{i} : a_{i}, b_{i})

is defined by

P (B_{i} \leq β_{i} : a_{i}, b_{i}) = \frac{Γ (a_{i} + b_{i})}{Γ (a_{i}) Γ (b_{i})} \int_{0}^{β_{i}} {B_{i}}^{a_{i} - 1} {(1 - B_{i})}^{b_{i} - 1} d B_{i} = I_{β_{i}} (a_{i}, b_{i}), 0 \leq β_{i} \leq 1; a_{i}, b_{i} > 0 .

The function

I_{β_{i}} (a_{i}, b_{i})

, also known as the incomplete beta function is calculated using S14CCF.

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

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

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

Majumder K L and Bhattacharjee G P (1973) Algorithm AS 63. The incomplete beta integral Appl. Statist. 22 409–411

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 probabilities are required. For

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

, for

i = 1, 2, \dots, \max (LTAIL, LBETA, LA, LB)

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

Constraint:

TAIL (j) ='L'

'U'

, for

j = 1, 2, \dots, LTAIL

3: LBETA – INTEGERInput

On entry: the length of the array BETA.

Constraint:

LBETA > 0

4: BETA(LBETA) – REAL (KIND=nag_wp) arrayInput

On entry:

β_{i}

, the value of the beta variate with

β_{i} = BETA (j)

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

Constraint:

0.0 \leq BETA (j) \leq 1.0

, for

j = 1, 2, \dots, LBETA

5: LA – INTEGERInput

On entry: the length of the array A.

Constraint:

LA > 0

6: A(LA) – REAL (KIND=nag_wp) arrayInput

On entry:

a_{i}

, the first parameter of the required beta distribution with

a_{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:

b_{i}

, the second parameter of the required beta distribution with

b_{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 (LTAIL, LBETA, LA, LB)

On exit:

p_{i}

, the probabilities for the beta distribution.

10: IVALID( $*$ ) – INTEGER arrayOutput

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

\max (LTAIL, LBETA, LA, LB)

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, $β_{i} < 0.0$ ,
or $β_{i} > 1.0$ .
$IVALID (i) = 3$: On entry, $a_{i} \leq 0.0$ ,
or $b_{i} \leq 0.0$ ,

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, 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: G01SEF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, at least one value of BETA, A, B or TAIL was invalid.
Check IVALID for more information.

$IFAIL = 2$: On entry, $array size = ⟨value⟩$ .
Constraint: $LTAIL > 0$ .

$IFAIL = 3$: On entry, $array size = ⟨value⟩$ .
Constraint: $LBETA > 0$ .

$IFAIL = 4$: 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 accuracy is limited by the error in the incomplete beta function. See Section 7 in S14CCF for further details.

8 Further Comments

None.

9 Example

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

NAG Library Routine DocumentG01SEF

+− 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

G01SEF