NAG Library Routine Document
g01fdf (inv_cdf_f)
1
Purpose
g01fdf returns the deviate associated with the given lower tail probability of the $F$ or varianceratio distribution with real degrees of freedom, via the routine name.
2
Specification
Fortran Interface
Real (Kind=nag_wp)  ::  g01fdf  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  p, df1, df2 

C Header Interface
#include nagmk26.h
double 
g01fdf_ (const double *p, const double *df1, const double *df2, Integer *ifail) 

3
Description
The deviate,
${f}_{p}$, associated with the lower tail probability,
$p$, of the
$F$distribution with degrees of freedom
${\nu}_{1}$ and
${\nu}_{2}$ is defined as the solution to
where
${\nu}_{1},{\nu}_{2}>0$;
$0\le {f}_{p}<\infty $.
The value of
${f}_{p}$ is computed by means of a transformation to a beta distribution,
${P}_{\beta}\left(B\le \beta :a,b\right)$:
and using a call to
g01fef.
For very large values of both
${\nu}_{1}$ and
${\nu}_{2}$, greater than
${10}^{5}$, a normal approximation is used. If only one of
${\nu}_{1}$ or
${\nu}_{2}$ is greater than
${10}^{5}$ then a
${\chi}^{2}$ approximation is used; see
Abramowitz and Stegun (1972).
4
References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
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 $F$distribution.
Constraint:
$0.0\le {\mathbf{p}}<1.0$.
 2: $\mathbf{df1}$ – Real (Kind=nag_wp)Input

On entry: the degrees of freedom of the numerator variance, ${\nu}_{1}$.
Constraint:
${\mathbf{df1}}>0.0$.
 3: $\mathbf{df2}$ – Real (Kind=nag_wp)Input

On entry: the degrees of freedom of the denominator variance, ${\nu}_{2}$.
Constraint:
${\mathbf{df2}}>0.0$.
 4: $\mathbf{ifail}$ – 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: g01fdf 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 ${\mathbf{ifail}}={\mathbf{1}}$, ${\mathbf{2}}$ or ${\mathbf{4}}$, then g01fdf returns $0.0$.
 ${\mathbf{ifail}}=1$

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

On entry, ${\mathbf{df1}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{df2}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{df1}}>0.0$ and ${\mathbf{df2}}>0.0$.
 ${\mathbf{ifail}}=3$

The solution has failed to converge. However, the result should be a reasonable approximation. Alternatively,
g01fef can be used with a suitable setting of the argument
tol.
 ${\mathbf{ifail}}=4$

The probability is too close to $0.0$ or $1.0$. The value of ${f}_{p}$ cannot be computed. This will only occur when the large sample approximations are used.
 ${\mathbf{ifail}}=99$
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 result should be accurate to five significant digits.
8
Parallelism and Performance
g01fdf is not threaded in any implementation.
For higher accuracy
g01fef can be used along with the transformations given in
Section 3.
10
Example
This example reads the lower tail probabilities for several $F$distributions, and calculates and prints the corresponding deviates until the end of data is reached.
10.1
Program Text
Program Text (g01fdfe.f90)
10.2
Program Data
Program Data (g01fdfe.d)
10.3
Program Results
Program Results (g01fdfe.r)