# NAG FL Interfaceg01edf (prob_​f)

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## 1Purpose

g01edf returns the probability for the lower or upper tail of the $F$ or variance-ratio distribution with real degrees of freedom.

## 2Specification

Fortran Interface
 Function g01edf ( tail, f, df1, df2,
 Real (Kind=nag_wp) :: g01edf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: f, df1, df2 Character (1), Intent (In) :: tail
C Header Interface
#include <nag.h>
 double g01edf_ (const char *tail, const double *f, const double *df1, const double *df2, Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01edf or nagf_stat_prob_f.

## 3Description

The lower tail probability for the $F$, or variance-ratio distribution, with ${\nu }_{1}$ and ${\nu }_{2}$ degrees of freedom, $P\left(F\le f:{\nu }_{1},{\nu }_{2}\right)$, is defined by:
 $P(F≤f:ν1,ν2)=ν1ν1/2ν2ν2/2 Γ ((ν1+ν2)/2) Γ(ν1/2) Γ(ν2/2) ∫0fF(ν1-2)/2(ν1F+ν2)-(ν1+ν2)/2dF,$
for ${\nu }_{1}$, ${\nu }_{2}>0$, $f\ge 0$.
The probability 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 g01eef.
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: $\mathbf{tail}$Character(1) Input
On entry: indicates whether an upper or lower tail probability is required.
${\mathbf{tail}}=\text{'L'}$
The lower tail probability is returned, i.e., $P\left(F\le f:{\nu }_{1},{\nu }_{2}\right)$.
${\mathbf{tail}}=\text{'U'}$
The upper tail probability is returned, i.e., $P\left(F\ge f:{\nu }_{1},{\nu }_{2}\right)$.
Constraint: ${\mathbf{tail}}=\text{'L'}$ or $\text{'U'}$.
2: $\mathbf{f}$Real (Kind=nag_wp) Input
On entry: $f$, the value of the $F$ variate.
Constraint: ${\mathbf{f}}\ge 0.0$.
3: $\mathbf{df1}$Real (Kind=nag_wp) Input
On entry: the degrees of freedom of the numerator variance, ${\nu }_{1}$.
Constraint: ${\mathbf{df1}}>0.0$.
4: $\mathbf{df2}$Real (Kind=nag_wp) Input
On entry: the degrees of freedom of the denominator variance, ${\nu }_{2}$.
Constraint: ${\mathbf{df2}}>0.0$.
5: $\mathbf{ifail}$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 g01edf may return useful information.
if ${\mathbf{ifail}}={\mathbf{1}}$, ${\mathbf{2}}$ or ${\mathbf{3}}$ on exit, then g01edf returns $0.0$.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{tail}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{tail}}=\text{'L'}$ or $\text{'U'}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{f}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{f}}\ge 0.0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{df1}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{df2}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df1}}>0.0$ and ${\mathbf{df2}}>0.0$.
${\mathbf{ifail}}=4$
The probability is too close to $0.0$ or $1.0$. f is too far out into the tails for the probability to be evaluated exactly. The result tends to approach $1.0$ if $f$ is large, or $0.0$ if $f$ is small. The result returned is a good approximation to the required solution.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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

Background information to multithreading can be found in the Multithreading documentation.
g01edf is not threaded in any implementation.

## 9Further Comments

For higher accuracy g01eef can be used along with the transformations given in Section 3.

## 10Example

This example reads values from, and degrees of freedom for, a number of $F$-distributions and computes the associated lower tail probabilities.

### 10.1Program Text

Program Text (g01edfe.f90)

### 10.2Program Data

Program Data (g01edfe.d)

### 10.3Program Results

Program Results (g01edfe.r)