NAG FL Interface
g01edf (prob_​f)

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1 Purpose

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

2 Specification

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 <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.

3 Description

The lower tail probability for the F, or variance-ratio distribution, with ν1 and ν2 degrees of freedom, P(Ff:ν1,ν2), is defined by:
P(Ff: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 ν1, ν2>0, f0.
The probability is computed by means of a transformation to a beta distribution, Pβ(Bβ:a,b):
P(Ff: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).

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: tail Character(1) Input
On entry: indicates whether an upper or lower tail probability is required.
The lower tail probability is returned, i.e., P(Ff:ν1,ν2).
The upper tail probability is returned, i.e., P(Ff: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 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 ifail0 on exit. 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).

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).
Errors or warnings detected by the routine:
Note: in some cases g01edf may return useful information.
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 7 in the Introduction to the NAG Library FL Interface for further information.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The result should be accurate to five significant digits.

8 Parallelism and Performance

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

9 Further Comments

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

10 Example

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

10.1 Program Text

Program Text (g01edfe.f90)

10.2 Program Data

Program Data (g01edfe.d)

10.3 Program Results

Program Results (g01edfe.r)