NAG Library Function Document

nag_prob_lin_chi_sq (g01jdc)

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

, then 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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
NE_REAL: On entry, $d = ⟨value⟩$ .
Constraint: $d \geq 0.0$ .
NE_REAL_ARRAY: On entry, all values of $rlam = d$ .
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 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 Documentnag_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

NAG Library Function Document

nag_prob_lin_chi_sq (g01jdc)