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 <nag.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)

The routine may be called by the names g01tdf or nagf_stat_inv_cdf_f_vector.

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: $ltail$ – Integer Input

On entry: the length of the array tail.

Constraint:

ltail > 0

2: $tail (ltail)$ – Character(1) array Input

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: $lp$ – Integer Input

On entry: the length of the array p.

Constraint:

lp > 0

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

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: $ldf1$ – Integer Input

On entry: the length of the array df1.

Constraint:

ldf1 > 0

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

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: $ldf2$ – Integer Input

On entry: the length of the array df2.

Constraint:

ldf2 > 0

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

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) array Output

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 array Output

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 $0.0$ or $1.0$ for the result to be computed. This will only occur when the large sample approximations are used.

11: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

−1

is recommended since useful values can be provided in some output arguments even when

ifail \neq 0

on exit. 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).

Errors or warnings detected by the routine:

Note: in some cases g01tdf may return useful information.

$ifail = 1$: 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 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The result should be accurate to five significant digits.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

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.

g01td: FL CL CPP AD PY MB

NAG FL Interfaceg01tdf (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

NAG FL Interface
g01tdf (inv_cdf_f_vector)