Magnus J R and Pesaran B (1993a) The evaluation of cumulants and moments of quadratic forms in Normal variables (CUM): Technical description Comput. Statist. 8 39–45

Magnus J R and Pesaran B (1993b) The evaluation of moments of quadratic forms and ratios of quadratic forms in Normal variables: Background, motivation and examples Comput. Statist. 8 47–55

Parameters

Compulsory Input Parameters

1: $a (lda, n)$ – double array: lda, the first dimension of the array, must satisfy the constraint $lda \geq n$ .
The $n$ by $n$ symmetric matrix $A$ . Only the lower triangle is referenced.
2: $sigma (ldsig, n)$ – double array: ldsig, the first dimension of the array, must satisfy the constraint $ldsig \geq n$ .
The $n$ by $n$ variance-covariance matrix $Σ$ . Only the lower triangle is referenced.

Constraint: the matrix $Σ$ must be positive definite.
3: $l$ – int64int32nag_int scalar: The required number of cumulants, and moments if specified.

Constraint: $1 \leq l \leq 12$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the arrays a, sigma and the second dimension of the arrays a, sigma. (An error is raised if these dimensions are not equal.)
$n$ , the dimension of the quadratic form.

Constraint: $n > 1$ .
2: $emu (:)$ – double array: The dimension of the array emu must be at least $n$ if $mean ='M'$ , and at least $1$ otherwise

If supplied, emu must contain the $n$ elements of the vector $μ$ .

Output Parameters

1: $rkum (l)$ – double array: The l cumulants of the quadratic form.
2: $rmom (:)$ – double array: The dimension of the array rmom will be $l$ if $mom ='M'$ and $1$ otherwise

If $mom ='M'$ , the l moments of the quadratic form.
3: $ifail$ – 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,	$n \leq 1$ ,
or	$l < 1$ ,
or	$l > 12$ ,
or	$lda < n$ ,
or	$ldsig < n$ ,
or	$mom \neq'C'$ or $'M'$ ,
or	$mean \neq'M'$ or $'Z'$ .

$ifail = 2$

On entry,

the matrix

Σ

is not positive definite.

$ifail = - 99$: 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

In a range of tests the accuracy was found to be a modest multiple of machine precision. See Magnus and Pesaran (1993b).

Further Comments

None.

Example

This example is given by Magnus and Pesaran (1993b) and considers the simple autoregression

y_{t} = β y_{t - 1} + u_{t}, t = 1, 2, \dots n,

where

\{u_{t}\}

is a sequence of independent Normal variables with mean zero and variance one, and

y_{0}

is known. The moments of the quadratic form

Q = \sum_{t = 2}^{n} y_{t} y_{t - 1}

are computed using nag_stat_moments_quad_form (g01na). The matrix

A

is given by:

\begin{array}{lcl} A (i + 1, i) & = & \frac{1}{2}, i = 1, 2, \dots n - 1; \\ A (i, j) & = & 0,   otherwise. \end{array}

The value of

Σ

can be computed using the relationships

var (y_{t}) = β^{2} var (y_{t - 1}) + 1

and

cov (y_{t} y_{t + k}) = β cov (y_{t} y_{t + k - 1})

for

k \geq 0

and

var (y_{1}) = 1

The values of

β

y_{0}

n

, and the number of moments required are read in and the moments and cumulants printed.

Open in the MATLAB editor: g01na_example

function g01na_example


fprintf('g01na example results\n\n');

% Problem parameters
n    = 10;
l    = int64(4);
beta = 0.8;
con  = 1.0;

% Simple autoregression setup
a            = zeros(n,n);
a(2:n,1:n-1) = 0.5*eye(n-1);
emu = zeros(n,1);
for j=1:n
  emu(j) = con*beta^j;
end
sigma = zeros(n,n);
sigma(1,1) = 1;
for j = 2:n
  sigma(j,j) = sigma(j-1,j-1)*beta^2 + 1;
end
for i = 1:n
  s = sigma(i,i);
  for j = i+1:n
    sigma(j,i) = s*beta^(j-i);
  end
end

[rkum, rmom, ifail] = g01na( ...
                             a, sigma, l, 'emu', emu);

% Display results
fprintf(' n = %3d, beta = %6.3f, con = %6.3f\n\n', n, beta, con);
fprintf('    Cumulants       Moments\n\n');
ival = double([1:l]');
fprintf('%3d%12.4e    %12.4e\n',[ival rkum rmom]');

g01na example results

 n =  10, beta =  0.800, con =  1.000

    Cumulants       Moments

  1  1.7517e+01      1.7517e+01
  2  3.5010e+02      6.5695e+02
  3  1.6091e+04      3.9865e+04
  4  1.1700e+06      3.4039e+06

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

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