NAG Library Routine Document
g01fef
(inv_cdf_beta)
1
Purpose
g01fef returns the deviate associated with the given lower tail probability of the beta distribution, via the routine name.
2
Specification
Fortran Interface
Real (Kind=nag_wp)  ::  g01fef  Integer, Intent (Inout)  :: 
ifail  Real (Kind=nag_wp), Intent (In)  :: 
p,
a,
b,
tol 

C Header Interface
#include nagmk26.h
double 
g01fef_ (
const double *p,
const double *a,
const double *b,
const double *tol,
Integer *ifail) 

3
Description
The deviate,
${\beta}_{p}$, associated with the lower tail probability,
$p$, of the beta distribution with parameters
$a$ and
$b$ is defined as the solution to
The algorithm is a modified version of the Newton–Raphson method, following closely that of
Cran et al. (1977).
An initial approximation,
${\beta}_{0}$, to
${\beta}_{p}$ is found (see
Cran et al. (1977)), and the Newton–Raphson iteration
where
$f\left(\beta \right)=P\left(B\le \beta :a,b\right)p$ is used, with modifications to ensure that
$\beta $ remains in the range
$\left(0,1\right)$.
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: $\mathbf{p}$ – Real (Kind=nag_wp)Input

On entry: $p$, the lower tail probability from the required beta distribution.
Constraint:
$0.0\le {\mathbf{p}}\le 1.0$.
 2: $\mathbf{a}$ – Real (Kind=nag_wp)Input

On entry: $a$, the first parameter of the required beta distribution.
Constraint:
$0.0<{\mathbf{a}}\le {10}^{6}$.
 3: $\mathbf{b}$ – Real (Kind=nag_wp)Input

On entry: $b$, the second parameter of the required beta distribution.
Constraint:
$0.0<{\mathbf{b}}\le {10}^{6}$.
 4: $\mathbf{tol}$ – Real (Kind=nag_wp)Input

On entry: the relative accuracy required by you in the result. If
g01fef is entered with
tol greater than or equal to
$1.0$ or less than
$10\times \mathit{machineprecision}$ (see
x02ajf), the value of
$10\times \mathit{machineprecision}$ is used instead.
 5: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ 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
${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is
$1$.
When the value $\mathbf{1}\text{ or}1$ 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).
6
Error 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: g01fef may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
If on exit ${\mathbf{ifail}}={\mathbf{1}}$ or ${\mathbf{2}}$, then g01fef returns $0.0$.
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{p}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{p}}\le 1.0$.
On entry, ${\mathbf{p}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{p}}\ge 0.0$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{a}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{b}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{a}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{b}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{a}}\le {10}^{6}$.
On entry, ${\mathbf{a}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{b}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{b}}>0.0$.
On entry, ${\mathbf{a}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{b}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{b}}\le {10}^{6}$.
 ${\mathbf{ifail}}=3$

The solution has failed to converge. However, the result should be a reasonable approximation. Requested accuracy not achieved when calculating beta probability. You should try setting
tol larger.
 ${\mathbf{ifail}}=4$

The requested accuracy has not been achieved. Use a larger value of
tol. There is doubt concerning the accuracy of the computed result.
$100$ iterations of the Newton–Raphson method have been performed without satisfying the accuracy criterion (see
Section 9). The result should be a reasonable approximation of the solution.
 ${\mathbf{ifail}}=99$
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.
 ${\mathbf{ifail}}=399$
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.
7
Accuracy
The required precision, given by
tol, should be achieved in most circumstances.
8
Parallelism and Performance
g01fef is not threaded in any implementation.
The typical timing will be several times that of
g01eef and will be very dependent on the input argument values. See
g01eef for further comments on timings.
10
Example
This example reads lower tail probabilities for several beta distributions and calculates and prints the corresponding deviates until the end of data is reached.
10.1
Program Text
Program Text (g01fefe.f90)
10.2
Program Data
Program Data (g01fefe.d)
10.3
Program Results
Program Results (g01fefe.r)