void	g01sec (Integer ltail, const Nag_TailProbability tail[], Integer lbeta, const double beta[], Integer la, const double a[], Integer lb, const double b[], double p[], Integer ivalid[], NagError *fail)

The function may be called by the names: g01sec, nag_stat_prob_beta_vector or nag_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 s14ccc.

The input arrays to this function 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: $tail [ltail]$ – const Nag_TailProbability Input

On entry: indicates whether a lower or upper tail probabilities are required. For

j = (i - 1) mod ltail

, for

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

$tail [j] = Nag_LowerTail$: The lower tail probability is returned, i.e., $p_{i} = P (B_{i} \leq β_{i} : a_{i}, b_{i})$ .
$tail [j] = Nag_UpperTail$: The upper tail probability is returned, i.e., $p_{i} = P (B_{i} \geq β_{i} : a_{i}, b_{i})$ .

Constraint:

tail [j - 1] = Nag_LowerTail

Nag_UpperTail

, for

j = 1, 2, \dots, ltail

3: $lbeta$ – Integer Input

On entry: the length of the array beta.

Constraint:

lbeta > 0

4: $beta [lbeta]$ – const double Input

On entry:

β_{i}

, the value of the beta variate with

β_{i} = beta [j]

j = (i - 1) mod lbeta

Constraint:

0.0 \leq beta [j - 1] \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: $a [la]$ – const double Input

On entry:

a_{i}

, the first parameter of the required beta distribution with

a_{i} = a [j]

j = (i - 1) mod la

Constraint:

a [j - 1] > 0.0

, for

j = 1, 2, \dots, la

7: $lb$ – Integer Input

On entry: the length of the array b.

Constraint:

lb > 0

8: $b [lb]$ – const double Input

On entry:

b_{i}

, the second parameter of the required beta distribution with

b_{i} = b [j]

j = (i - 1) mod lb

Constraint:

b [j - 1] > 0.0

, for

j = 1, 2, \dots, lb

9: $p [\dim]$ – double Output

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

\max (ltail, lbeta, la, lb)

On exit:

p_{i}

, the probabilities for the beta distribution.

10: $ivalid [\dim]$ – Integer Output

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

\max (ltail, lbeta, la, lb)

On exit:

ivalid [i - 1]

indicates any errors with the input arguments, with

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: On entry, invalid value supplied in tail when calculating $p_{i}$ .
$ivalid [i - 1] = 2$: On entry, $β_{i} < 0.0$ , or, $β_{i} > 1.0$ .
$ivalid [i - 1] = 3$: On entry, $a_{i} \leq 0.0$ , or, $b_{i} \leq 0.0$ .

11: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE: On entry, $array size = ⟨ value ⟩$ .
Constraint: $la > 0$ .

On entry, $array size = ⟨ value ⟩$ .
Constraint: $lb > 0$ .

On entry, $array size = ⟨ value ⟩$ .
Constraint: $lbeta > 0$ .

On entry, $array size = ⟨ value ⟩$ .
Constraint: $ltail > 0$ .
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID: On entry, at least one value of beta, a, b or tail was invalid.
Check ivalid for more information.

7 Accuracy

The accuracy is limited by the error in the incomplete beta function. See Section 7 in s14ccc for further details.

8 Parallelism and Performance

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

g01sec 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 CL Interfaceg01sec (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 CL Interface
g01sec (prob_beta_vector)