# NAG FL Interfaceg01fdf (inv_​cdf_​f)

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## 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 <nag.h>
 double g01fdf_ (const double *p, const double *df1, const double *df2, Integer *ifail)
The routine may be called by the names g01fdf or nagf_stat_inv_cdf_f.

## 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
 $P ( F ≤ fp : ν1 ,ν2) = p = ν 1 12 ν1 ν 2 12 ν2 Γ ( ν1 + ν2 2 ) Γ ( ν1 2 ) Γ ( ν2 2 ) ∫ 0 fp F 12 (ν1-2) (ν2+ν1F) -12 (ν1+ν2) dF ,$
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, ${P}_{\beta }\left(B\le \beta :a,b\right)$:
 $P(F≤f:ν1,ν2)=Pβ (B≤ν1f ν1f+ν2 :ν1/2,ν2/2)$
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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $-1$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ 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).

## 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).
Errors or warnings detected by the routine:
Note: in some cases g01fdf may return useful information.
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 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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)