NAG FL Interface
g01fdf (inv_​cdf_​f)

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1 Purpose

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

2 Specification

Fortran Interface
Function g01fdf ( p, df1, df2, ifail)
Real (Kind=nag_wp) :: g01fdf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: p, df1, df2
C Header Interface
#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.

3 Description

The deviate, fp, associated with the lower tail probability, p, of the F-distribution with degrees of freedom ν1 and ν2 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 ν1,ν2>0; 0fp<.
The value of fp 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 g01fef.
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: p Real (Kind=nag_wp) Input
On entry: p, the lower tail probability from the required F-distribution.
Constraint: 0.0p<1.0.
2: df1 Real (Kind=nag_wp) Input
On entry: the degrees of freedom of the numerator variance, ν1.
Constraint: df1>0.0.
3: df2 Real (Kind=nag_wp) Input
On entry: the degrees of freedom of the denominator variance, ν2.
Constraint: df2>0.0.
4: 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 g01fdf may return useful information.
If on exit ifail=1, 2 or 4, then g01fdf returns 0.0.
On entry, p=value.
Constraint: p<1.0.
On entry, p=value.
Constraint: p0.0.
On entry, df1=value and df2=value.
Constraint: df1>0.0 and df2>0.0.
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.
The probability is too close to 0.0 or 1.0. The value of fp cannot be computed. This will only occur when the large sample approximations are used.
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

g01fdf is not threaded in any implementation.

9 Further Comments

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

10 Example

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.

10.1 Program Text

Program Text (g01fdfe.f90)

10.2 Program Data

Program Data (g01fdfe.d)

10.3 Program Results

Program Results (g01fdfe.r)