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 is used to compute the value of the integral otherwise nag_1d_quad_gen_1 (d01sjc) is used.

Pan's procedure can only be used if the

λ_{i}^{*}

are sufficiently distinct; nag_prob_lin_chi_sq (g01jdc) 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: $method$ – Nag_LCCMethodInput

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

$method = Nag_LCCPan$: Pan's method is used.
$method = Nag_LCCImhof$: Imhof's method is used.
$method = Nag_LCCDefault$: 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 = Nag_LCCPan

Nag_LCCImhof

Nag_LCCDefault

2: $n$ – IntegerInput

On entry:

n

, the number of independent standard Normal variates, (central

χ^{2}

variates).

Constraint:

n \geq 1

3: $rlam [n]$ – const doubleInput

On entry: the weights,

λ_{i}

, for

i = 1, 2, \dots, n

, of the central

χ^{2}

variables.

Constraint:

rlam [i - 1] \neq d

for at least one

i

. If

method = Nag_LCCPan

, the

λ_{i}^{*}

must be at least

1 %

distinct; see Section 9, for

i = 1, 2, \dots, n

4: $d$ – doubleInput

On entry:

d

, the multiplier of the central

χ^{2}

variables.

Constraint:

d \geq 0.0

5: $c$ – doubleInput

On entry:

c

, the value of the constant.

6: $prob$ – double *Output

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

χ^{2}

variables.

7: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL: On entry, $d = 〈value〉$ .
Constraint: $d \geq 0.0$ .
NE_REAL_ARRAY: On entry, $rlam [i - 1] = d$ for all values of $i$ , for $i = 1, 2, \dots, n$ .
NE_REAL_ARRAY_ENUM: On entry, $method = Nag_LCCPan$ but two successive values of $λ *$ were not $1$ percent distinct.

7

Accuracy

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

8

Parallelism and Performance

nag_prob_lin_chi_sq (g01jdc) 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 function. 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. nag_prob_lin_chi_sq (g01jdc) 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 nag_prob_lin_non_central_chi_sq (g01jcc) may be used. If the probabilities required are for the Durbin–Watson test, then the bounds for the probabilities are given by nag_prob_durbin_watson (g01epc).

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 Function Document

nag_prob_lin_chi_sq (g01jdc)

▸▿ 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