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

p: Type: System..::..Double
On entry: $p$ , the lower tail probability from the required $F$ -distribution.

Constraint: $0.0 \leq p < 1.0$ .

df1: Type: System..::..Double
On entry: the degrees of freedom of the numerator variance, $ν_{1}$ .

Constraint: $df1 > 0.0$ .

df2: Type: System..::..Double
On entry: the degrees of freedom of the denominator variance, $ν_{2}$ .

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]).

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,

f_{p}

, associated with the lower tail probability,

p

, of the

F

-distribution with degrees of freedom

ν_{1}

and

ν_{2}

is defined as the solution to

P (F \leq f_{p} : ν_{1}, ν_{2}) = p = \frac{ν_{1}^{\frac{1}{2} ν_{1}} ν_{2}^{\frac{1}{2} ν_{2}} Γ (\frac{ν_{1} + ν_{2}}{2})}{Γ (\frac{ν_{1}}{2}) Γ (\frac{ν_{2}}{2})} \int_{0}^{f_{p}} F^{\frac{1}{2} (ν_{1} - 2)} {(ν_{2} + ν_{1} F)}^{- \frac{1}{2} (ν_{1} + ν_{2})} d F,

where

ν_{1}, ν_{2} > 0

;

0 \leq f_{p} < \infty

The value of

f_{p}

is computed by means of a transformation to a beta distribution,

P_{β} (B \leq β : a, b)

P (F \leq f : ν_{1}, ν_{2}) = P_{β} (B \leq \frac{ν_{1} f}{ν_{1} f + ν_{2}} : ν_{1} / 2, ν_{2} / 2)

and using a call to g01fe.

For very large values of both

ν_{1}

and

ν_{2}

, greater than

10^{5}

, a normal approximation is used. If only one of

ν_{1}

ν_{2}

is greater than

10^{5}

then a

χ^{2}

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

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

4

, then g01fd returns

0.0

$ifail = 1$

On entry,	$p < 0.0$ ,
or	$p \geq 1.0$ .

$ifail = 2$

On entry,	$df1 \leq 0.0$ ,
or	$df2 \leq 0.0$ .

$ifail = 3$: 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 $f_{p}$ to be computed. This will only occur when the large sample approximations are used.