g01gd returns the probability associated with the lower tail of the noncentral

F

or variance-ratio distribution.

Syntax

C#
public static double g01gd( double f, double df1, double df2, double rlamda, double tol, int maxit, out int ifail )

Visual Basic
Public Shared Function g01gd ( _ f As Double, _ df1 As Double, _ df2 As Double, _ rlamda As Double, _ tol As Double, _ maxit As Integer, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double g01gd( double f, double df1, double df2, double rlamda, double tol, int maxit, [OutAttribute] int% ifail )

F#
static member g01gd : f : float * df1 : float * df2 : float * rlamda : float * tol : float * maxit : int * ifail : int byref -> float

Parameters

f: Type: System..::..Double
On entry: $f$ , the deviate from the noncentral $F$ -distribution.

Constraint: $f > 0.0$ .

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

Constraint: $0.0 < df1 \leq 10^{6}$ .

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

Constraint: $df2 > 0.0$ .

rlamda: Type: System..::..Double
On entry: $λ$ , the noncentrality parameter.

Constraint: $0.0 \leq rlamda \leq - 2.0 \log (U)$ where $U$ is the safe range parameter as defined by x02am.

tol: Type: System..::..Double
On entry: the relative accuracy required by you in the results. If g01gd is entered with tol greater than or equal to $1.0$ or less than $10 \times machine precision$ (see x02aj), then the value of $10 \times machine precision$ is used instead.

maxit: Type: System..::..Int32
On entry: the maximum number of iterations to be used.
Suggested value: $500$ . See g01gc and g01ge for further details.

Constraint: $maxit \geq 1$ .

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

g01gd returns the probability associated with the lower tail of the noncentral

F

or variance-ratio distribution.

Description

The lower tail probability of the noncentral

F

-distribution with

ν_{1}

and

ν_{2}

degrees of freedom and noncentrality parameter

λ

P (F \leq f : ν_{1}, ν_{2}; λ)

, is defined by

P (F \leq f : ν_{1}, ν_{2}; λ) = \int_{0}^{x} p (F : ν_{1}, ν_{2}; λ) d F,

where

P (F : ν_{1}, ν_{2}; λ) = \sum_{j = 0}^{\infty} e^{- λ / 2} \frac{{(λ / 2)}^{j}}{j!} \times \frac{{(ν_{1} + 2 j)}^{(ν_{1} + 2 j) / 2} ν_{2}^{ν_{2} / 2}}{B ((ν_{1} + 2 j) / 2, ν_{2} / 2)}

\times u^{(ν_{1} + 2 j - 2) / 2} {[ν_{2} + (ν_{1} + 2 j) u]}^{- (ν_{1} + 2 j + ν_{2}) / 2}

and

B (\cdot, \cdot)

is the beta function.

The probability is computed by means of a transformation to a noncentral beta distribution:

P (F \leq f : ν_{1}, ν_{2}; λ) = P_{β} (X \leq x : a, b; λ),

where

x = \frac{ν_{1} f}{ν_{1} f + ν_{2}}

and

P_{β} (X \leq x : a, b; λ)

is the lower tail probability integral of the noncentral beta distribution with parameters

a

b

, and

λ

ν_{2}

is very large, greater than

10^{6}

, then a

χ^{2}

approximation is used.

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Error Indicators and Warnings

Note: g01gd 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

3

, then g01gd returns

0.0

$ifail = 1$

On entry,	$df1 \leq 0.0$ ,
or	$df1 > 10^{6}$ ,
or	$df2 \leq 0.0$ ,
or	$f \leq 0.0$ ,
or	$rlamda < 0.0$ ,
or	$maxit < 1$ ,
or	$rlamda > - 2.0 \log (U)$ , where $U =$ safe range parameter as defined by x02am.

$ifail = 2$: The solution has failed to converge in maxit iterations. You should try a larger value of maxit or tol.

$ifail = 3$: The required probability cannot be computed accurately. This may happen if the result would be very close to $0.0$ or $1.0$ . Alternatively the values of df1 and f may be too large. In the latter case you could try using a normal approximation; see Abramowitz and Stegun (1972).

$ifail = 4$: The required accuracy was not achieved when calculating the initial value of the central $F$ (or $χ^{2}$ ) probability. You should try a larger value of tol. If the $χ^{2}$ approximation is being used then g01gd returns zero otherwise the value returned should be an approximation to the correct value.

$ifail = -9000$: An error occured, see message report.

Accuracy

The relative accuracy should be as specified by tol. For further details see g01gc and g01ge.

Parallelism and Performance

None.

Further Comments

When both

ν_{1}

and

ν_{2}

are large a Normal approximation may be used and when only

ν_{1}

is large a

χ^{2}

approximation may be used. In both cases

λ

is required to be of the same order as

ν_{1}

. See Abramowitz and Stegun (1972) for further details.

Example

This example reads values from, and degrees of freedom for,

F

-distributions, computes the lower tail probabilities and prints all these values until the end of data is reached.

Example program (C#): g01gde.cs

Example program data: g01gde.d

Example program results: g01gde.r