g01ed returns the probability for the lower or upper tail of the

F

or variance-ratio distribution with real degrees of freedom.

Syntax

C#
public static double g01ed( string tail, double f, double df1, double df2, out int ifail )

Visual Basic
Public Shared Function g01ed ( _ tail As String, _ f As Double, _ df1 As Double, _ df2 As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double g01ed( String^ tail, double f, double df1, double df2, [OutAttribute] int% ifail )

F#
static member g01ed : tail : string * f : float * df1 : float * df2 : float * ifail : int byref -> float

Parameters

tail

Type: System..::..String

On entry: indicates whether an upper or lower tail probability is required.

$tail = "L"$: The lower tail probability is returned, i.e., $P (F \leq f : ν_{1}, ν_{2})$ .
$tail = "U"$: The upper tail probability is returned, i.e., $P (F \geq f : ν_{1}, ν_{2})$ .

Constraint:

tail = "L"

"U"

f: Type: System..::..Double
On entry: $f$ , the value of the $F$ variate.

Constraint: $f \geq 0.0$ .

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]).

Return Value

g01ed returns the probability for the lower or upper tail of the

F

or variance-ratio distribution with real degrees of freedom.

Description

The lower tail probability for the

F

, or variance-ratio distribution, with

ν_{1}

and

ν_{2}

degrees of freedom,

P (F \leq f : ν_{1}, ν_{2})

, is defined by:

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

for

ν_{1}

ν_{2} > 0

f \geq 0

The probability 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 g01ee.

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).

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: g01ed may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

ifail = 1

2

3

on exit, then g01ed returns

0.0

$ifail = 1$

On entry,

tail \neq "L"

"U"

$ifail = 2$

On entry,

f < 0.0

$ifail = 3$

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

$ifail = 4$: 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.