NAG Library Routine Document

Integer, Intent (In)	::	ltail, lp, ldf1, ldf2
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(*)
Real (Kind=nag_wp), Intent (In)	::	p(lp), df1(ldf1), df2(ldf2)
Real (Kind=nag_wp), Intent (Out)	::	f(*)
Character (1), Intent (In)	::	tail(ltail)

C Header Interface

#include <nagmk26.h>

void	g01tdf_ (const Integer ltail, const char tail[], const Integer lp, const double p[], const Integer ldf1, const double df1[], const Integer ldf2, const double df2[], double f[], Integer ivalid[], Integer *ifail, const Charlen length_tail)

3

Description

The deviate,

f_{p_{i}}

, associated with the lower tail probability,

p_{i}

, of the

F

-distribution with degrees of freedom

u_{i}

and

v_{i}

is defined as the solution to

P (F_{i} \leq f_{p_{i}} : u_{i}, v_{i}) = p_{i} = \frac{u_{i}^{\frac{1}{2} u_{i}} v_{i}^{\frac{1}{2} v_{i}} Γ (\frac{u_{i} + v_{i}}{2})}{Γ (\frac{u_{i}}{2}) Γ (\frac{v_{i}}{2})} \int_{0}^{f_{p_{i}}} {F_{i}}^{\frac{1}{2} (u_{i} - 2)} {(v_{i} + u_{i} F_{i})}^{- \frac{1}{2} (u_{i} + v_{i})} d F_{i},

where

u_{i}, v_{i} > 0

;

0 \leq f_{p_{i}} < \infty

The value of

f_{p_{i}}

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

P_{i β_{i}} (B_{i} \leq β_{i} : a_{i}, b_{i})

P (F_{i} \leq f_{p_{i}} : u_{i}, v_{i}) = P_{i β_{i}} (B_{i} \leq \frac{u_{i} f_{p_{i}}}{u_{i} f_{p_{i}} + v_{i}} : u_{i} / 2, v_{i} / 2)

and using a call to g01tef.

For very large values of both

u_{i}

and

v_{i}

, greater than

10^{5}

, a Normal approximation is used. If only one of

u_{i}

v_{i}

is greater than

10^{5}

then a

χ^{2}

approximation is used; see Abramowitz and Stegun (1972).

The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

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: $ltail$ – IntegerInput

On entry: the length of the array tail.

Constraint:

ltail > 0

2: $tail (ltail)$ – Character(1) arrayInput

On entry: indicates which tail the supplied probabilities represent. For

j = ((i - 1) mod ltail) + 1

, for

i = 1, 2, \dots, \max (ltail, lp, ldf1, ldf2)

$tail (j) ='L'$: The lower tail probability, i.e., $p_{i} = P (F_{i} \leq f_{p_{i}} : u_{i}, v_{i})$ .
$tail (j) ='U'$: The upper tail probability, i.e., $p_{i} = P (F_{i} \geq f_{p_{i}} : u_{i}, v_{i})$ .

Constraint:

tail (j) ='L'

'U'

, for

j = 1, 2, \dots, ltail

3: $lp$ – IntegerInput

On entry: the length of the array p.

Constraint:

lp > 0

4: $p (lp)$ – Real (Kind=nag_wp) arrayInput

On entry:

p_{i}

, the probability of the required

F

-distribution as defined by tail with

p_{i} = p (j)

j = ((i - 1) mod lp) + 1

Constraints:

if $tail (k) ='L'$ , $0.0 \leq p (j) < 1.0$ ;
otherwise $0.0 < p (j) \leq 1.0$ .

Where

k = (i - 1) mod ltail + 1

and

j = (i - 1) mod lp + 1

5: $ldf1$ – IntegerInput

On entry: the length of the array df1.

Constraint:

ldf1 > 0

6: $df1 (ldf1)$ – Real (Kind=nag_wp) arrayInput

On entry:

u_{i}

, the degrees of freedom of the numerator variance with

u_{i} = df1 (j)

j = ((i - 1) mod ldf1) + 1

Constraint:

df1 (j) > 0.0

, for

j = 1, 2, \dots, ldf1

7: $ldf2$ – IntegerInput

On entry: the length of the array df2.

Constraint:

ldf2 > 0

8: $df2 (ldf2)$ – Real (Kind=nag_wp) arrayInput

On entry:

v_{i}

, the degrees of freedom of the denominator variance with

v_{i} = df2 (j)

j = ((i - 1) mod ldf2) + 1

Constraint:

df2 (j) > 0.0

, for

j = 1, 2, \dots, ldf2

9: $f (*)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array f must be at least

\max (ltail, lp, ldf1, ldf2)

On exit:

f_{p_{i}}

, the deviates for the

F

-distribution.

10: $ivalid (*)$ – Integer arrayOutput

Note: the dimension of the array ivalid must be at least

\max (ltail, lp, ldf1, ldf2)

On exit:

ivalid (i)

indicates any errors with the input arguments, with

$ivalid (i) = 0$

No error.

$ivalid (i) = 1$

On entry,

invalid value supplied in tail when calculating

f_{p_{i}}

$ivalid (i) = 2$

On entry,

invalid value for

p_{i}

$ivalid (i) = 3$

On entry,	$u_{i} \leq 0.0$ ,
or	$v_{i} \leq 0.0$ .

$ivalid (i) = 4$

The solution has not converged. The result should still be a reasonable approximation to the solution.

$ivalid (i) = 5$

The value of

p_{i}

is too close to

0.0

1.0

for the result to be computed. This will only occur when the large sample approximations are used.

11: $ifail$ – 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: g01tdf may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$ifail = 1$: On entry, at least one value of tail, p, df1, df2 was invalid, or the solution failed to converge.
Check ivalid for more information.

$ifail = 2$: On entry, $array size = 〈value〉$ .
Constraint: $ltail > 0$ .

$ifail = 3$: On entry, $array size = 〈value〉$ .
Constraint: $lp > 0$ .

$ifail = 4$: On entry, $array size = 〈value〉$ .
Constraint: $ldf1 > 0$ .

$ifail = 5$: On entry, $array size = 〈value〉$ .
Constraint: $ldf2 > 0$ .

$ifail = - 99$: 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

g01tdf is not threaded in any implementation.

9

Further Comments

For higher accuracy g01tef 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.

NAG Library Routine Document

g01tdf (inv_cdf_f_vector)

▸▿ 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

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