Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	rlam(n), d, c
Real (Kind=nag_wp), Intent (Out)	::	prob, work(n+1)
Character (1), Intent (In)	::	method

C Header Interface

#include nagmk26.h

void	g01jdf_ ( const char method, const Integer n, const double rlam[], const double d, const double c, double prob, double work[], Integer ifail, const Charlen length_method)

3

Description

Let

u_{1}, u_{2}, \dots, u_{n}

be independent Normal variables with mean zero and unit variance, so that

u_{1}^{2}, u_{2}^{2}, \dots, u_{n}^{2}

have independent

χ^{2}

-distributions with unit degrees of freedom. g01jdf evaluates the probability that

λ_{1} u_{1}^{2} + λ_{2} u_{2}^{2} + \dots + λ_{n} u_{n}^{2} < d (u_{1}^{2} + u_{2}^{2} + \dots + u_{n}^{2}) + c .

c = 0.0

this is equivalent to the probability that

\frac{λ_{1} u_{1}^{2} + λ_{2} u_{2}^{2} + \dots + λ_{n} u_{n}^{2}}{u_{1}^{2} + u_{2}^{2} + \dots + u_{n}^{2}} < d .

Alternatively let

λ_{i}^{*} = λ_{i} - d,   ​ i = 1, 2, \dots, n,

then g01jdf returns the probability that

λ_{1}^{*} u_{1}^{2} + λ_{2}^{*} u_{2}^{2} + \dots + λ_{n}^{*} u_{n}^{2} < c .

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 9. 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

Arguments

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'

, the

λ_{i}^{*}

must be at least

1 %

distinct; see Section 9, 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 argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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, $d = 〈value〉$ .
Constraint: $d \geq 0.0$ .

On entry, $method = 〈value〉$ .
Constraint: $method ='P'$ , $'I'$ or $'D'$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

$ifail = 2$: On entry, $rlam (i) = d$ for all values of $i$ , for $i = 1, 2, \dots, n$ .

$ifail = 3$: On entry, $method ='P'$ but two successive values of $λ *$ were not $1$ percent distinct.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

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

8

Parallelism and Performance

g01jdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

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.

10

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 Document

g01jdf (prob_chisq_lincomb)

▸▿ 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

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