NAG C Library Function Document

1Purpose

nag_deviates_beta_vector (g01tec) returns a number of deviates associated with given probabilities of the beta distribution.

2Specification

 #include #include
 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)

3Description

The deviate, ${\beta }_{{p}_{i}}$, 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
 $P Bi ≤ βpi :ai,bi = pi = Γ ai + bi Γ ai Γ bi ∫ 0 βpi Bi ai-1 1-Bi bi-1 d Bi , 0 ≤ β pi ≤ 1 ; ​ ai , bi > 0 .$
The algorithm is a modified version of the Newton–Raphson method, following closely that of Cran et al. (1977).
An initial approximation, ${\beta }_{i0}$, to ${\beta }_{{p}_{i}}$ is found (see Cran et al. (1977)), and the Newton–Raphson iteration
 $βk = βk-1 - fi βk-1 fi′ βk-1 ,$
where ${f}_{i}\left({\beta }_{k}\right)=P\left({B}_{i}\le {\beta }_{k}:{a}_{i},{b}_{i}\right)-{p}_{i}$ is used, with modifications to ensure that ${\beta }_{k}$ remains in the range $\left(0,1\right)$.
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.

4References

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

5Arguments

1:    $\mathbf{ltail}$IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:    $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbabilityInput
On entry: indicates which tail the supplied probabilities represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability, i.e., ${p}_{i}=P\left({B}_{i}\le {\beta }_{{p}_{i}}:{a}_{i},{b}_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability, i.e., ${p}_{i}=P\left({B}_{i}\ge {\beta }_{{p}_{i}}:{a}_{i},{b}_{i}\right)$.
Constraint: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$ or $\mathrm{Nag_UpperTail}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:    $\mathbf{lp}$IntegerInput
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4:    $\mathbf{p}\left[{\mathbf{lp}}\right]$const doubleInput
On entry: ${p}_{i}$, the probability of the required beta distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left[j\right]$, .
Constraint: $0.0\le {\mathbf{p}}\left[\mathit{j}-1\right]\le 1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lp}}$.
5:    $\mathbf{la}$IntegerInput
On entry: the length of the array a.
Constraint: ${\mathbf{la}}>0$.
6:    $\mathbf{a}\left[{\mathbf{la}}\right]$const doubleInput
On entry: ${a}_{i}$, the first parameter of the required beta distribution with ${a}_{i}={\mathbf{a}}\left[j\right]$, .
Constraint: $0.0<{\mathbf{a}}\left[\mathit{j}-1\right]\le {10}^{6}$, for $\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
7:    $\mathbf{lb}$IntegerInput
On entry: the length of the array b.
Constraint: ${\mathbf{lb}}>0$.
8:    $\mathbf{b}\left[{\mathbf{lb}}\right]$const doubleInput
On entry: ${b}_{i}$, the second parameter of the required beta distribution with ${b}_{i}={\mathbf{b}}\left[j\right]$, .
Constraint: $0.0<{\mathbf{b}}\left[\mathit{j}-1\right]\le {10}^{6}$, for $\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.
9:    $\mathbf{tol}$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  (see nag_machine_precision (X02AJC)), the value of  is used instead.
10:  $\mathbf{beta}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array beta must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$.
On exit: ${\beta }_{{p}_{i}}$, the deviates for the beta distribution.
11:  $\mathbf{ivalid}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$.
On exit: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
 On entry, invalid value supplied in tail when calculating ${\beta }_{{p}_{i}}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
 On entry, ${p}_{i}<0.0$, or ${p}_{i}>1.0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
 On entry, ${a}_{i}\le 0.0$, or ${a}_{i}>{10}^{6}$, or ${b}_{i}\le 0.0$, or ${b}_{i}>{10}^{6}$.
${\mathbf{ivalid}}\left[i-1\right]=4$
The solution has not converged but the result should be a reasonable approximation to the solution.
${\mathbf{ivalid}}\left[i-1\right]=5$
Requested accuracy not achieved when calculating the beta probability. The result should be a reasonable approximation to the correct solution.
12:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error 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, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{la}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lb}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lp}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
On entry, argument $〈\mathit{\text{value}}〉$ 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 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.

7Accuracy

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

8Parallelism and Performance

nag_deviates_beta_vector (g01tec) is not threaded in any implementation.

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.

10Example

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

10.1Program Text

Program Text (g01tece.c)

10.2Program Data

Program Data (g01tece.d)

10.3Program Results

Program Results (g01tece.r)