NAG Library Routine Document

G01JDF

Two methods are available. One due to Pan (1964) (see Farebrother (1980)) makes use of series approximations. The other method due to Imhof (1961) reduces the problem to a one-dimensional integral. If

n \geq 6

then a non-adaptive method described in D01BDF is used to compute the value of the integral otherwise D01AJF is used.

Pan's procedure can only be used if the

λ_{i}^{*}

are sufficiently distinct; G01JDF requires the

λ_{i}^{*}

to be at least

1 %

distinct; see Section 8. If the

λ_{i}^{*}

are at least

1 %

distinct and

n \leq 60

, then Pan's procedure is recommended; otherwise Imhof's procedure is recommended.

4 References

Farebrother R W (1980) Algorithm AS 153. Pan's procedure for the tail probabilities of the Durbin–Watson statistic Appl. Statist. 29 224–227

Imhof J P (1961) Computing the distribution of quadratic forms in Normal variables Biometrika 48 419–426

Pan Jie–Jian (1964) Distributions of the noncircular serial correlation coefficients Shuxue Jinzhan 7 328–337

5 Parameters

1: METHOD – CHARACTER(1)Input

On entry: indicates whether Pan's, Imhof's or an appropriately selected procedure is to be used.

$METHOD ='P'$: Pan's method is used.
$METHOD ='I'$: Imhof's method is used.
$METHOD ='D'$: Pan's method is used if $λ_{i}^{*}$ , for $i = 1, 2, \dots, n$ are at least $1 %$ distinct and $n \leq 60$ ; otherwise Imhof's method is used.

Constraint:

METHOD ='P'

'I'

'D'

2: N – INTEGERInput

On entry:

n

, the number of independent standard Normal variates, (central

χ^{2}

variates).

Constraint:

N \geq 1

3: RLAM(N) – REAL (KIND=nag_wp) arrayInput

On entry: the weights,

λ_{i}

, for

i = 1, 2, \dots, n

, of the central

χ^{2}

variables.

Constraint:

RLAM (i) \neq D

for at least one

i

. If

METHOD ='P'

, then the

λ_{i}^{*}

must be at least

1 %

distinct; see Section 8, for

i = 1, 2, \dots, n

4: D – REAL (KIND=nag_wp)Input

On entry:

d

, the multiplier of the central

χ^{2}

variables.

Constraint:

D \geq 0.0

5: C – REAL (KIND=nag_wp)Input

On entry:

c

, the value of the constant.

6: PROB – REAL (KIND=nag_wp)Output

On exit: the lower tail probability for the linear combination of central

χ^{2}

variables.

7: WORK( $N + 1$ ) – REAL (KIND=nag_wp) arrayWorkspace

8: 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, if you are not familiar with this parameter, the recommended value is

0

. 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:

$IFAIL = 1$

On entry,	$N < 1$ ,
or	$D < 0.0$ ,
or	$METHOD \neq'P'$ , $'I'$ or $'D'$ .

$IFAIL = 2$: On entry, $RLAM (i) = D$ for all values of $i$ , for $i = 1, 2, \dots, n$ .

$IFAIL = 3$: On entry, $METHOD ='P'$ yet two successive values of the ordered $λ_{i}^{*}$ , for $i = 1, 2, \dots, n$ , were not at least $1 %$ distinct.

7 Accuracy

On successful exit at least four decimal places of accuracy should be achieved.

8 Further Comments

Pan's procedure can only work if the

λ_{i}^{*}

are sufficiently distinct. G01JDF uses the check

|w_{j} - w_{j - 1}| \geq 0.01 \times \max (|w_{j}|, |w_{j - 1}|)

, where the

w_{j}

are the ordered nonzero values of

λ_{i}^{*}

For the situation when all the

λ_{i}

are positive G01JCF may be used. If the probabilities required are for the Durbin–Watson test, then the bounds for the probabilities are given by G01EPF.

9 Example

For

n = 10

, the choice of method, values of

c

and

d

and the

λ_{i}

are input and the probabilities computed and printed.

NAG Library Routine DocumentG01JDF

+− 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

G01JDF