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 nag_quad_1d_fin_smooth (d01bd) is used to compute the value of the integral otherwise nag_quad_1d_fin_bad (d01aj) is used.

Pan's procedure can only be used if the

λ_{i}^{*}

are sufficiently distinct; nag_stat_prob_chisq_lincomb (g01jd) requires the

λ_{i}^{*}

to be at least

1 %

distinct; see Further Comments. 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.

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

Parameters

Compulsory Input Parameters

1: $rlam (n)$ – double array: 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 Further Comments, for $i = 1, 2, \dots, n$ .
2: $d$ – double scalar: $d$ , the multiplier of the central $χ^{2}$ variables.

Constraint: $d \geq 0.0$ .
3: $c$ – double scalar: $c$ , the value of the constant.

Optional Input Parameters

1: $method$ – string (length ≥ 1)

Default:

'D'

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: $n$ – int64int32nag_int scalar

Default: the dimension of the array rlam.

n

, the number of independent standard Normal variates, (central

χ^{2}

variates).

Constraint:

n \geq 1

Output Parameters

1: $prob$ – double scalar: The lower tail probability for the linear combination of central $χ^{2}$ variables.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

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

Accuracy

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

Further Comments

Pan's procedure can only work if the

λ_{i}^{*}

are sufficiently distinct. nag_stat_prob_chisq_lincomb (g01jd) 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_stat_prob_chisq_noncentral_lincomb (g01jc) may be used. If the probabilities required are for the Durbin–Watson test, then the bounds for the probabilities are given by nag_stat_prob_durbin_watson (g01ep).

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.

Open in the MATLAB editor: g01jd_example

function g01jd_example


fprintf('g01jd example results\n\n');

method = 'P';
rlam = [-9 -7 -5 -3 -1  2  4  6  8  10];
d = 1;
c = 0;

[p, ifail] = g01jd( ...
		    rlam, d, c, 'method', method);

fprintf('lambda = ');
fprintf('%6.1f',rlam);
fprintf('\n     d = %6.1f\n', d);
fprintf('     c = %6.1f\n\n', c);
fprintf('  prob = %9.4f\n', p);

g01jd example results

lambda =   -9.0  -7.0  -5.0  -3.0  -1.0   2.0   4.0   6.0   8.0  10.0
     d =    1.0
     c =    0.0

  prob =    0.5749

PDF version (NAG web site, 64-bit version, 64-bit version)

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