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

C Header Interface

#include <nag.h>

void	g01sef_ (const Integer ltail, const char tail[], const Integer lbeta, const double beta[], 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 g01sef or nagf_stat_prob_beta_vector.

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 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

NIST Digital Library of Mathematical Functions

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 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 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: $lbeta$ – Integer Input

On entry: the length of the array beta.

Constraint:

lbeta > 0

4: $beta (lbeta)$ – Real (Kind=nag_wp) array Input

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: $la$ – Integer Input

On entry: the length of the array a.

Constraint:

la > 0

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

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: $lb$ – Integer Input

On entry: the length of the array b.

Constraint:

lb > 0

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

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) array Output

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 array Output

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: $ifail$ – 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

−1

is recommended since useful values can be provided in some output arguments even when

ifail \neq 0

on exit. 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:

Note: in some cases g01sef may return useful information.

$ifail = 1$: 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: $lb > 0$ .

$ifail = - 99$: 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 accuracy is limited by the error in the incomplete beta function. See Section 7 in s14ccf for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g01sef is not threaded in any implementation.

9 Further Comments

None.

10 Example

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

g01se: FL CL CPP AD PY MB

NAG FL Interfaceg01sef (prob_​beta_​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
g01sef (prob_beta_vector)