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NAG Toolbox: nag_stat_prob_chisq_lincomb (g01jd)

Purpose

nag_stat_prob_chisq_lincomb (g01jd) calculates the lower tail probability for a linear combination of (central) ${\chi }^{2}$ variables.

Syntax

[prob, ifail] = g01jd(rlam, d, c, 'method', method, 'n', n)
[prob, ifail] = nag_stat_prob_chisq_lincomb(rlam, d, c, 'method', method, 'n', n)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 23: method was made optional (default 'D')

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 ${\chi }^{2}$-distributions with unit degrees of freedom. nag_stat_prob_chisq_lincomb (g01jd) evaluates the probability that
 $λ1u12+λ2u22+⋯+λnun2
If $c=0.0$ this is equivalent to the probability that
 $λ1u12+λ2u22+⋯+λnun2 u12+u22+⋯+un2
Alternatively let
 $λi*=λi-d, ​ i= 1,2,…,n,$
then nag_stat_prob_chisq_lincomb (g01jd) returns the probability that
 $λ1*u12+λ2*u22+⋯+λn*un2
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 $n\ge 6$ 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 ${\lambda }_{i}^{*}$ are sufficiently distinct; nag_stat_prob_chisq_lincomb (g01jd) requires the ${\lambda }_{i}^{*}$ to be at least $1%$ distinct; see Further Comments. If the ${\lambda }_{i}^{*}$ are at least $1%$ distinct and $n\le 60$, 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:     $\mathrm{rlam}\left({\mathbf{n}}\right)$ – double array
The weights, ${\lambda }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, of the central ${\chi }^{2}$ variables.
Constraint: ${\mathbf{rlam}}\left(\mathit{i}\right)\ne {\mathbf{d}}$ for at least one $\mathit{i}$. If ${\mathbf{method}}=\text{'P'}$, then the ${\lambda }_{\mathit{i}}^{*}$ must be at least $1%$ distinct; see Further Comments, for $\mathit{i}=1,2,\dots ,n$.
2:     $\mathrm{d}$ – double scalar
$d$, the multiplier of the central ${\chi }^{2}$ variables.
Constraint: ${\mathbf{d}}\ge 0.0$.
3:     $\mathrm{c}$ – double scalar
$c$, the value of the constant.

Optional Input Parameters

1:     $\mathrm{method}$ – string (length ≥ 1)
Default: $\text{'D'}$
Indicates whether Pan's, Imhof's or an appropriately selected procedure is to be used.
${\mathbf{method}}=\text{'P'}$
Pan's method is used.
${\mathbf{method}}=\text{'I'}$
Imhof's method is used.
${\mathbf{method}}=\text{'D'}$
Pan's method is used if ${\lambda }_{\mathit{i}}^{*}$, for $\mathit{i}=1,2,\dots ,n$ are at least $1%$ distinct and $n\le 60$; otherwise Imhof's method is used.
Constraint: ${\mathbf{method}}=\text{'P'}$, $\text{'I'}$ or $\text{'D'}$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array rlam.
$n$, the number of independent standard Normal variates, (central ${\chi }^{2}$ variates).
Constraint: ${\mathbf{n}}\ge 1$.

Output Parameters

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

Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<1$, or ${\mathbf{d}}<0.0$, or ${\mathbf{method}}\ne \text{'P'}$, $\text{'I'}$ or $\text{'D'}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{rlam}}\left(\mathit{i}\right)={\mathbf{d}}$ for all values of $\mathit{i}$, for $\mathit{i}=1,2,\dots ,n$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{method}}=\text{'P'}$ yet two successive values of the ordered ${\lambda }_{\mathit{i}}^{*}$, for $\mathit{i}=1,2,\dots ,n$, were not at least $1%$ distinct.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Accuracy

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

Pan's procedure can only work if the ${\lambda }_{i}^{*}$ are sufficiently distinct. nag_stat_prob_chisq_lincomb (g01jd) uses the check $\left|{w}_{j}-{w}_{j-1}\right|\ge 0.01×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\left|{w}_{j}\right|,\left|{w}_{j-1}\right|\right)$, where the ${w}_{j}$ are the ordered nonzero values of ${\lambda }_{i}^{*}$.
For the situation when all the ${\lambda }_{i}$ 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 ${\lambda }_{i}$ are input and the probabilities computed and printed.
```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
```

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