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

C Header Interface

#include <nag.h>

void	g01sdf_ (const Integer ltail, const char tail[], const Integer lf, const double f[], const Integer ldf1, const double df1[], const Integer ldf2, const double df2[], double p[], Integer ivalid[], Integer *ifail, const Charlen length_tail)

The routine may be called by the names g01sdf or nagf_stat_prob_f_vector.

3 Description

The lower tail probability for the

F

, or variance-ratio, distribution with

u_{i}

and

v_{i}

degrees of freedom,

P (F_{i} \leq f_{i} : u_{i}, v_{i})

, is defined by:

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

for

u_{i}

v_{i} > 0

f_{i} \geq 0

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

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

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

and using a call to g01eef.

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 whether the lower or upper tail probabilities are required. For

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

, for

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

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

Constraint:

tail (j) ='L'

'U'

, for

j = 1, 2, \dots, ltail

3: $lf$ – Integer Input

On entry: the length of the array f.

Constraint:

lf > 0

4: $f (lf)$ – Real (Kind=nag_wp) array Input

On entry:

f_{i}

, the value of the

F

variate with

f_{i} = f (j)

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

Constraint:

f (j) \geq 0.0

, for

j = 1, 2, \dots, lf

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: $p (*)$ – Real (Kind=nag_wp) array Output

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

\max (ltail, lf, ldf1, ldf2)

On exit:

p_{i}

, the probabilities for the

F

-distribution.

10: $ivalid (*)$ – Integer array Output

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

\max (ltail, lf, 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 $p_{i}$ .
$ivalid (i) = 2$: On entry, $f_{i} < 0.0$ .
$ivalid (i) = 3$: On entry, $u_{i} \leq 0.0$ , or, $v_{i} \leq 0.0$ .
$ivalid (i) = 4$: The solution has failed to converge. The result returned should represent an approximation to the solution.

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 g01sdf may return useful information.

$ifail = 1$: On entry, at least one value of f, df1, df2 or tail 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: $lf > 0$ .

$ifail = 4$: 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

g01sdf is not threaded in any implementation.

9 Further Comments

For higher accuracy g01sef can be used along with the transformations given in Section 3.

10 Example

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

F

-distributions and computes the associated lower tail probabilities.

g01sd: FL CL CPP AD

NAG FL Interfaceg01sdf (prob_​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
g01sdf (prob_f_vector)