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

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 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

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: $tail [ltail]$ – const Nag_TailProbabilityInput

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

j = (i - 1) mod ltail

, for

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

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

Constraint:

tail [j - 1] = Nag_LowerTail

Nag_UpperTail

, for

j = 1, 2, \dots, ltail

3: $lp$ – IntegerInput

On entry: the length of the array p.

Constraint:

lp > 0

4: $p [lp]$ – const doubleInput

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

Constraint:

0.0 \leq p [j - 1] \leq 1.0

, for

j = 1, 2, \dots, lp

5: $la$ – IntegerInput

On entry: the length of the array a.

Constraint:

la > 0

6: $a [la]$ – const doubleInput

On entry:

a_{i}

, the first parameter of the required beta distribution with

a_{i} = a [j]

j = (i - 1) mod la

Constraint:

0.0 < a [j - 1] \leq 10^{6}

, for

j = 1, 2, \dots, la

7: $lb$ – IntegerInput

On entry: the length of the array b.

Constraint:

lb > 0

8: $b [lb]$ – const doubleInput

On entry:

b_{i}

, the second parameter of the required beta distribution with

b_{i} = b [j]

j = (i - 1) mod lb

Constraint:

0.0 < b [j - 1] \leq 10^{6}

, for

j = 1, 2, \dots, lb

9: $tol$ – doubleInput

On entry: the relative accuracy required by you in the results. If nag_deviates_beta_vector (g01tec) is entered with tol greater than or equal to

1.0

or less than

10 \times machine precision

(see nag_machine_precision (X02AJC)), then the value of

10 \times machine precision

is used instead.

10: $beta [\dim]$ – doubleOutput

Note: the dimension, dim, 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: $ivalid [\dim]$ – IntegerOutput

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

\max (ltail, lp, 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

β_{p_{i}}

$ivalid [i - 1] = 2$

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

$ivalid [i - 1] = 3$

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

$ivalid [i - 1] = 4$

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

$ivalid [i - 1] = 5$

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

12: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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: $lp > 0$ .

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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID: 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.

7 Accuracy

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

8 Parallelism and Performance

nag_deviates_beta_vector (g01tec) is not threaded in any implementation.

9 Further Comments

The typical timing will be several times that of nag_prob_beta_vector (g01sec) and will be very dependent on the input argument values. See nag_prob_beta_vector (g01sec) 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 Function Documentnag_deviates_beta_vector (g01tec)

▸▿ 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 Library Function Document

nag_deviates_beta_vector (g01tec)