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 (Declaration)
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 probability from the required F-distribution.

Constraint: 0.0≤p<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 (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

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 + ν1 F -12 ν1 + ν2 dF ,

where ν1,ν2>0; 0≤fp<∞.

The value of fp is computed by means of a transformation to a beta distribution, PβB≤β:a,b:

PF≤f:ν1,ν2=Pβ B≤ν1f ν1f+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=1, 2 or 4, then g01fd returns 0.0.

ifail=1: On entry, p<0.0,
or p≥1.0.

ifail=2: On entry, df1≤0.0,
or df2≤0.0.

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.