g01fd returns the deviate associated with the given lower tail probability of the F or variance-ratio distribution with real degrees of freedom.

Syntax

C#
public static double g01fd(
	double p,
	double df1,
	double df2,
	out int ifail
)
Visual Basic
Public Shared Function g01fd ( _
	p As Double, _
	df1 As Double, _
	df2 As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
public:
static double g01fd(
	double p, 
	double df1, 
	double df2, 
	[OutAttribute] int% ifail
)
F#
static member g01fd : 
        p : float * 
        df1 : float * 
        df2 : float * 
        ifail : int byref -> float 

Parameters

p
Type: System..::..Double
On entry: p, the lower tail probability from the required F-distribution.
Constraint: 0.0p<1.0.
df1
Type: System..::..Double
On entry: the degrees of freedom of the numerator variance, ν1.
Constraint: df1>0.0.
df2
Type: System..::..Double
On entry: the degrees of freedom of the denominator variance, ν2.
Constraint: df2>0.0.
ifail
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01fd returns the deviate associated with the given lower tail probability of the F or variance-ratio distribution with real degrees of freedom.

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
PFfp:ν1,ν2=p=ν112ν1ν212ν2Γν1+ν22Γν12Γν220fpF12ν1-2ν2+ν1F-12ν1+ν2dF,
where ν1,ν2>0; 0fp<.
The value of fp is computed by means of a transformation to a beta distribution, PβBβ:a,b:
PFf:ν1,ν2=PβBν1fν1f+ν2:ν1/2,ν2/2
and using a call to g01fe.
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).

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

Error Indicators and Warnings

Note: g01fd may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
If on exit ifail=12 or 4, then g01fd returns 0.0.
ifail=1
On entry,p<0.0,
orp1.0.
ifail=2
On entry,df10.0,
ordf20.0.
ifail=3
The solution has not converged. The result should still be a reasonable approximation to the solution. Alternatively, g01fe can be used with a suitable setting of the parameter tol.
ifail=4
The value of p is too close to 0 or 1 for the value of fp to be computed. This will only occur when the large sample approximations are used.
ifail=-9000
An error occured, see message report.

Accuracy

The result should be accurate to five significant digits.

Parallelism and Performance

None.

Further Comments

For higher accuracy g01fe can be used along with the transformations given in [Description].

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.

Example program (C#): g01fde.cs

Example program data: g01fde.d

Example program results: g01fde.r

See Also