# NAG Library Routine Document

## 1Purpose

g01fdf returns the deviate associated with the given lower tail probability of the $F$ or variance-ratio distribution with real degrees of freedom.

## 2Specification

Fortran Interface
 Function g01fdf ( p, df1, df2,
 Real (Kind=nag_wp) :: g01fdf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: p, df1, df2
#include <nagmk26.h>
 double g01fdf_ (const double *p, const double *df1, const double *df2, Integer *ifail)

## 3Description

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
 $P F ≤ fp : ν1 ,ν2 = p = ν 1 12 ν1 ν 2 12 ν2 Γ ν1 + ν2 2 Γ ν1 2 Γ ν2 2 ∫ 0 fp F 12 ν1-2 ν2 + ν1 F -12 ν1 + ν2 dF ,$
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, ${P}_{\beta }\left(B\le \beta :a,b\right)$:
 $PF≤f:ν1,ν2=Pβ B≤ν1f ν1f+ν2 :ν1/2,ν2/2$
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).

## 4References

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

## 5Arguments

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 $0$, . 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: 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, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}<1.0$.
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}\ge 0.0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{df1}}=〈\mathit{\text{value}}〉$ and ${\mathbf{df2}}=〈\mathit{\text{value}}〉$.
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.
${\mathbf{ifail}}=-99$
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 result should be accurate to five significant digits.

## 8Parallelism and Performance

g01fdf is not threaded in any implementation.

For higher accuracy g01fef can be used along with the transformations given in Section 3.

## 10Example

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.

### 10.1Program Text

Program Text (g01fdfe.f90)

### 10.2Program Data

Program Data (g01fdfe.d)

### 10.3Program Results

Program Results (g01fdfe.r)