# NAG Library Routine Document

## 1Purpose

g01tef returns a number of deviates associated with given probabilities of the beta distribution.

## 2Specification

Fortran Interface
 Subroutine g01tef ( tail, lp, p, la, a, lb, b, tol, beta,
 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)
#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)

## 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 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.

## 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:     $\mathbf{tail}\left({\mathbf{ltail}}\right)$ – Character(1) arrayInput
On entry: indicates which tail the supplied probabilities represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability, i.e., ${p}_{i}=P\left({B}_{i}\le {\beta }_{{p}_{i}}:{a}_{i},{b}_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability, i.e., ${p}_{i}=P\left({B}_{i}\ge {\beta }_{{p}_{i}}:{a}_{i},{b}_{i}\right)$.
Constraint: ${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$ or $\text{'U'}$, 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:     $\mathbf{p}\left({\mathbf{lp}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${p}_{i}$, the probability of the required beta distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left(j\right)$, .
Constraint: $0.0\le {\mathbf{p}}\left(\mathit{j}\right)\le 1.0$, 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:     $\mathbf{a}\left({\mathbf{la}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${a}_{i}$, the first parameter of the required beta distribution with ${a}_{i}={\mathbf{a}}\left(j\right)$, .
Constraint: $0.0<{\mathbf{a}}\left(\mathit{j}\right)\le {10}^{6}$, 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:     $\mathbf{b}\left({\mathbf{lb}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${b}_{i}$, the second parameter of the required beta distribution with ${b}_{i}={\mathbf{b}}\left(j\right)$, .
Constraint: $0.0<{\mathbf{b}}\left(\mathit{j}\right)\le {10}^{6}$, for $\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.
9:     $\mathbf{tol}$ – 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  (see x02ajf), the value of  is used instead.
10:   $\mathbf{beta}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension 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:   $\mathbf{ivalid}\left(*\right)$ – Integer arrayOutput
Note: the dimension 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: ${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
 On entry, invalid value supplied in tail when calculating ${\beta }_{{p}_{i}}$.
${\mathbf{ivalid}}\left(i\right)=2$
 On entry, ${p}_{i}<0.0$, or ${p}_{i}>1.0$.
${\mathbf{ivalid}}\left(i\right)=3$
 On entry, ${a}_{i}\le 0.0$, or ${a}_{i}>{10}^{6}$, or ${b}_{i}\le 0.0$, or ${b}_{i}>{10}^{6}$.
${\mathbf{ivalid}}\left(i\right)=4$
The solution has not converged but the result should be a reasonable approximation to the solution.
${\mathbf{ivalid}}\left(i\right)=5$
Requested accuracy not achieved when calculating the beta probability. The result should be a reasonable approximation to the correct solution.
12:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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 ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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:
${\mathbf{ifail}}=1$
On entry, at least one value of tail, p, a, or b was invalid, or the solution failed to converge.
${\mathbf{ifail}}=2$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lp}}>0$.
${\mathbf{ifail}}=4$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{la}}>0$.
${\mathbf{ifail}}=5$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lb}}>0$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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

## 8Parallelism and Performance

g01tef is not threaded in any implementation.

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.

## 10Example

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

### 10.1Program Text

Program Text (g01tefe.f90)

### 10.2Program Data

Program Data (g01tefe.d)

### 10.3Program Results

Program Results (g01tefe.r)