NAG Library Routine Document

G01FDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G01FDF returns the deviate associated with the given lower tail probability of the

F

or variance-ratio distribution with real degrees of freedom, via the routine name.

2 Specification

FUNCTION G01FDF (

P, DF1, DF2, IFAIL)

REAL (KIND=nag_wp) G01FDF

INTEGER	IFAIL
REAL (KIND=nag_wp)	P, DF1, DF2

3 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 G01FEF.

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).

4 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

5 Parameters

1: P – REAL (KIND=nag_wp)Input: On entry: $p$ , the lower tail probability from the required $F$ -distribution.
Constraint: $0.0 \leq P < 1.0$ .
2: DF1 – REAL (KIND=nag_wp)Input: On entry: the degrees of freedom of the numerator variance, $ν_{1}$ .

Constraint: $DF1 > 0.0$ .
3: DF2 – REAL (KIND=nag_wp)Input: On entry: the degrees of freedom of the denominator variance, $ν_{2}$ .

Constraint: $DF2 > 0.0$ .
4: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if $IFAIL \neq 0$ on exit, the recommended value is $- 1$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Note: G01FDF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

If on exit

IFAIL = 1

2

4

, then G01FDF 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, G01FEF 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.

7 Accuracy

The result should be accurate to five significant digits.

8 Further Comments

For higher accuracy G01FEF can be used along with the transformations given in Section 3.

9 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.

NAG Library Routine DocumentG01FDF