NAG Library Routine Document

G01TDF

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

G01TDF returns a number of deviates associated with given probabilities of the

F

or variance-ratio distribution with real degrees of freedom.

2 Specification

SUBROUTINE G01TDF ( LTAIL, TAIL, LP, P, LDF1, DF1, LDF2, DF2, F, IVALID, IFAIL)

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

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

CHARACTER(1) TAIL(LTAIL)

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 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 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 $0.0$ or $1.0$ 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 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: 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: $LDF2 > 0$ .

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

7 Accuracy

The result should be accurate to five significant digits.

8 Further Comments

For higher accuracy G01TEF can be used along with the transformations given in Section 3.

9 Example

This example reads the lower tail probabilities for several

F

-distributions, and calculates and prints the corresponding deviates.

NAG Library Routine DocumentG01TDF

+− 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

G01TDF