nag_moments_quad_form (g01nac) (PDF version)
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g01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_moments_quad_form (g01nac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_moments_quad_form (g01nac) computes the cumulants and moments of quadratic forms in Normal variates.

2  Specification

#include <nag.h>
#include <nagg01.h>
void  nag_moments_quad_form (Nag_OrderType order, Nag_SelectMoments mom, Nag_IncludeMean mean, Integer n, const double a[], Integer pda, const double emu[], const double sigma[], Integer pdsig, Integer l, double rkum[], double rmom[], NagError *fail)

3  Description

Let x have an n-dimensional multivariate Normal distribution with mean μ and variance-covariance matrix Σ. Then for a symmetric matrix A, nag_moments_quad_form (g01nac) computes up to the first 12 moments and cumulants of the quadratic form Q=xTAx. The sth moment (about the origin) is defined as
EQs,
where E denotes expectation. The sth moment of Q can also be found as the coefficient of ts/s! in the expansion of EeQt. The sth cumulant is defined as the coefficient of ts/s! in the expansion of logEeQt.
The function is based on the function CUM written by Magnus and Pesaran (1993a) and based on the theory given by Magnus (1978), Magnus (1979) and Magnus (1986).

4  References

Magnus J R (1978) The moments of products of quadratic forms in Normal variables Statist. Neerlandica 32 201–210
Magnus J R (1979) The expectation of products of quadratic forms in Normal variables: the practice Statist. Neerlandica 33 131–136
Magnus J R (1986) The exact moments of a ratio of quadratic forms in Normal variables Ann. Économ. Statist. 4 95–109
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

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     momNag_SelectMomentsInput
On entry: indicates if moments are computed in addition to cumulants.
mom=Nag_CumulantsOnly
Only cumulants are computed.
mom=Nag_ComputeMoments
Moments are computed in addition to cumulants.
Constraint: mom=Nag_CumulantsOnly or Nag_ComputeMoments.
3:     meanNag_IncludeMeanInput
On entry: indicates if the mean, μ, is zero.
mean=Nag_MeanZero
μ is zero.
mean=Nag_MeanInclude
The value of μ is supplied in emu.
Constraint: mean=Nag_MeanZero or Nag_MeanInclude.
4:     nIntegerInput
On entry: n, the dimension of the quadratic form.
Constraint: n>1.
5:     a[dim]const doubleInput
Note: the dimension, dim, of the array a must be at least pda×n.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the n by n symmetric matrix A. Only the lower triangle is referenced.
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdan.
7:     emu[dim]const doubleInput
Note: the dimension, dim, of the array emu must be at least
  • n when mean=Nag_MeanInclude;
  • 1 otherwise.
On entry: if mean=Nag_MeanInclude, emu must contain the n elements of the vector μ.
If mean=Nag_MeanZero, emu is not referenced.
8:     sigma[dim]const doubleInput
Note: the dimension, dim, of the array sigma must be at least pdsig×n.
The i,jth element of the matrix is stored in
  • sigma[j-1×pdsig+i-1] when order=Nag_ColMajor;
  • sigma[i-1×pdsig+j-1] when order=Nag_RowMajor.
On entry: the n by n variance-covariance matrix Σ. Only the lower triangle is referenced.
Constraint: the matrix Σ must be positive definite.
9:     pdsigIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array sigma.
Constraint: pdsign.
10:   lIntegerInput
On entry: the required number of cumulants, and moments if specified.
Constraint: 1l12.
11:   rkum[l]doubleOutput
On exit: the l cumulants of the quadratic form.
12:   rmom[dim]doubleOutput
Note: the dimension, dim, of the array rmom must be at least
  • l when mom=Nag_ComputeMoments;
  • 1 otherwise.
On exit: if mom=Nag_ComputeMoments, the l moments of the quadratic form.
13:   failNagError *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, l=value.
Constraint: l12.
On entry, l=value.
Constraint: l1.
On entry, n=value.
Constraint: n>1.
On entry, pda=value.
Constraint: pda>0.
On entry, pdsig=value.
Constraint: pdsig>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdan.
On entry, pdsig=value and n=value.
Constraint: pdsign.
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_POS_DEF
On entry, sigma is not positive definite.

7  Accuracy

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

8  Further Comments

None.

9  Example

This example is given by Magnus and Pesaran (1993b) and considers the simple autoregression
yt=βyt-1+ut,   t=1,2,n,
where ut is a sequence of independent Normal variables with mean zero and variance one, and y0 is known. The moments of the quadratic form
Q=t=2nytyt-1
are computed using nag_moments_quad_form (g01nac). The matrix A is given by:
Ai+1,i = 12,  i=1,2,n-1; Ai,j = 0,   otherwise.
The value of Σ can be computed using the relationships
varyt=β2varyt-1+1
and
covytyt+k=β covytyt+k- 1
for k0 and vary1=1.
The values of β, y0, n, and the number of moments required are read in and the moments and cumulants printed.

9.1  Program Text

Program Text (g01nace.c)

9.2  Program Data

Program Data (g01nace.d)

9.3  Program Results

Program Results (g01nace.r)


nag_moments_quad_form (g01nac) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012