NAG Library Routine Document

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

C Header Interface

#include <nagmk26.h>

void	g01tef_ (const Integer ltail, const char tail[], const Integer lp, const double p[], const Integer la, const double a[], const Integer lb, const double b[], const double tol, double beta[], Integer ivalid[], Integer ifail, const Charlen length_tail)

3

Description

The deviate,

β_{p_{i}}

, associated with the lower tail probability,

p_{i}

, of the beta distribution with parameters

a_{i}

and

b_{i}

is defined as the solution to

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

The algorithm is a modified version of the Newton–Raphson method, following closely that of Cran et al. (1977).

An initial approximation,

β_{i 0}

, to

β_{p_{i}}

is found (see Cran et al. (1977)), and the Newton–Raphson iteration

β_{k} = β_{k - 1} - \frac{f_{i} (β_{k - 1})}{{f_{i}}^{'} (β_{k - 1})},

where

f_{i} (β_{k}) = P (B_{i} \leq β_{k} : a_{i}, b_{i}) - p_{i}

is used, with modifications to ensure that

β_{k}

remains in the range

(0, 1)

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

Cran G W, Martin K J and Thomas G E (1977) Algorithm AS 109. Inverse of the incomplete beta function ratio Appl. Statist. 26 111–114

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

5

Arguments

1: $ltail$ – IntegerInput

On entry: the length of the array tail.

Constraint:

ltail > 0

2: $tail (ltail)$ – Character(1) arrayInput

On entry: indicates which tail the supplied probabilities represent. For

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

, for

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

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

Constraint:

tail (j) ='L'

'U'

, for

j = 1, 2, \dots, ltail

3: $lp$ – IntegerInput

On entry: the length of the array p.

Constraint:

lp > 0

4: $p (lp)$ – Real (Kind=nag_wp) arrayInput

On entry:

p_{i}

, the probability of the required beta distribution as defined by tail with

p_{i} = p (j)

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

Constraint:

0.0 \leq p (j) \leq 1.0

, for

j = 1, 2, \dots, lp

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:

0.0 < a (j) \leq 10^{6}

, 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:

0.0 < b (j) \leq 10^{6}

, for

j = 1, 2, \dots, lb

9: $tol$ – Real (Kind=nag_wp)Input

On entry: the relative accuracy required by you in the results. If g01tef is entered with tol greater than or equal to

1.0

or less than

10 \times machine precision

(see x02ajf), the value of

10 \times machine precision

is used instead.

10: $beta (*)$ – Real (Kind=nag_wp) arrayOutput

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

\max (ltail, lp, la, lb)

On exit:

β_{p_{i}}

, the deviates for the beta distribution.

11: $ivalid (*)$ – Integer arrayOutput

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

\max (ltail, lp, 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

β_{p_{i}}

$ivalid (i) = 2$

On entry,	$p_{i} < 0.0$ ,
or	$p_{i} > 1.0$ .

$ivalid (i) = 3$

On entry,	$a_{i} \leq 0.0$ ,
or	$a_{i} > 10^{6}$ ,
or	$b_{i} \leq 0.0$ ,
or	$b_{i} > 10^{6}$ .

$ivalid (i) = 4$

The solution has not converged but the result should be a reasonable approximation to the solution.

$ivalid (i) = 5$

Requested accuracy not achieved when calculating the beta probability. The result should be a reasonable approximation to the correct solution.

12: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 arguments 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: g01tef 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 tail, p, a, or b was invalid, or the solution failed to converge.
Check ivalid for more information.

$ifail = 2$: On entry, $array size = 〈value〉$ .
Constraint: $ltail > 0$ .

$ifail = 3$: On entry, $array size = 〈value〉$ .
Constraint: $lp > 0$ .

$ifail = 4$: 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 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The required precision, given by tol, should be achieved in most circumstances.

8

Parallelism and Performance

g01tef is not threaded in any implementation.

9

Further Comments

The typical timing will be several times that of g01sef and will be very dependent on the input argument values. See g01sef for further comments on timings.

10

Example

This example reads lower tail probabilities for several beta distributions and calculates and prints the corresponding deviates.

NAG Library Routine Document

g01tef (inv_cdf_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

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