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NAG Library Routine Document

g01fdf (inv_cdf_f)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 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

Fortran Interface

Function g01fdf (

p, df1, df2, ifail)

Real (Kind=nag_wp)	::	g01fdf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	p, df1, df2

C Header Interface

#include nagmk26.h

double

g01fdf_ ( const double *p, const double *df1, const double *df2, Integer *ifail)

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

Arguments

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 argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 arguments 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, $p = 〈value〉$ .
Constraint: $p \geq 0.0$ .

On entry, $p = 〈value〉$ .
Constraint: $p < 1.0$ .

$ifail = 2$: On entry, $df1 = 〈value〉$ and $df2 = 〈value〉$ .
Constraint: $df1 > 0.0$ and $df2 > 0.0$ .

$ifail = 3$: The solution has failed to converge. However, the result should be a reasonable approximation. Alternatively, g01fef can be used with a suitable setting of the argument tol.

$ifail = 4$: The probability is too close to $0.0$ or $1.0$ . The value of $f_{p}$ cannot be computed. This will only occur when the large sample approximations are used.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The result should be accurate to five significant digits.

8

Parallelism and Performance

g01fdf is not threaded in any implementation.

9

Further Comments

For higher accuracy g01fef can be used along with the transformations given in Section 3.

10

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 Document

g01fdf (inv_cdf_f)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results