NAG Library Routine Document

G01SDF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− 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

G01SDF returns a number of lower or upper tail probabilities for the

F

or variance-ratio distribution with real degrees of freedom.

2 Specification

SUBROUTINE G01SDF ( LTAIL, TAIL, LF, F, LDF1, DF1, LDF2, DF2, P, IVALID, IFAIL)

INTEGER LTAIL, LF, LDF1, LDF2, IVALID(*), IFAIL

REAL (KIND=nag_wp) F(LF), DF1(LDF1), DF2(LDF2), P(*)

CHARACTER(1) TAIL(LTAIL)

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 parameters 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 Parameters

1: LTAIL – INTEGERInput

On entry: the length of the array TAIL.

Constraint:

LTAIL > 0

2: TAIL(LTAIL) – CHARACTER(1) arrayInput

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 – INTEGERInput

On entry: the length of the array F.

Constraint:

LF > 0

4: F(LF) – REAL (KIND=nag_wp) arrayInput

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 – 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: P( $*$ ) – REAL (KIND=nag_wp) arrayOutput

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 arrayOutput

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 – 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: G01SDF 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 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: $LDF2 > 0$ .

$IFAIL = - 999$: Dynamic memory allocation failed.

7 Accuracy

The result should be accurate to five significant digits.

8 Further Comments

For higher accuracy G01SEF can be used along with the transformations given in Section 3.

9 Example

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

F

-distributions and computes the associated lower tail probabilities.

NAG Library Routine DocumentG01SDF

+− 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

NAG Library Routine Document

G01SDF