NAG Library Routine Document

g01edf (prob_f)


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


Fortran Interface
Function g01edf ( tail, f, df1, df2, ifail)
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 <nagmk26.h>
double  g01edf_ (const char *tail, const double *f, const double *df1, const double *df2, Integer *ifail, const Charlen length_tail)


The lower tail probability for the F, or variance-ratio distribution, with ν1 and ν2 degrees of freedom, PFf:ν1,ν2, is defined by:
PFf:ν1,ν2=ν1ν1/2ν2ν2/2 Γ ν1+ν2/2 Γν1/2 Γν2/2 0fFν1-2/2ν1F+ν2- ν1+ν2/2dF,  
for ν1, ν2>0, f0.
The probability is computed by means of a transformation to a beta distribution, PβBβ:a,b:
PFf:ν1,ν2=Pβ Bν1f ν1f+ν2 :ν1/2,ν2/2  
and using a call to g01eef.
For very large values of both ν1 and ν2, greater than 105, a normal approximation is used. If only one of ν1 or ν2 is greater than 105 then a χ2 approximation is used, see Abramowitz and Stegun (1972).


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


1:     tail – Character(1)Input
On entry: indicates whether an upper or lower tail probability is required.
The lower tail probability is returned, i.e., PFf:ν1,ν2.
The upper tail probability is returned, i.e., PFf:ν1,ν2.
Constraint: tail='L' or 'U'.
2:     f – Real (Kind=nag_wp)Input
On entry: f, the value of the F variate.
Constraint: f0.0.
3:     df1 – Real (Kind=nag_wp)Input
On entry: the degrees of freedom of the numerator variance, ν1.
Constraint: df1>0.0.
4:     df2 – Real (Kind=nag_wp)Input
On entry: the degrees of freedom of the denominator variance, ν2.
Constraint: df2>0.0.
5:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 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 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 ifail0 on exit, the recommended value is -1. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Note: g01edf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
If ifail=1, 2 or 3 on exit, then g01edf returns 0.0.
On entry, tail=value.
Constraint: tail='L' or 'U'.
On entry, f=value.
Constraint: f0.0.
On entry, df1=value and df2=value.
Constraint: df1>0.0 and df2>0.0.
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.
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.
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.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


The result should be accurate to five significant digits.

Parallelism and Performance

g01edf is not threaded in any implementation.

Further Comments

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


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

Program Text

Program Text (g01edfe.f90)

Program Data

Program Data (g01edfe.d)

Program Results

Program Results (g01edfe.r)