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## 1Purpose

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

## 2Specification

 #include
 void g01nac (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)
The function may be called by the names: g01nac, nag_stat_moments_quad_form or nag_moments_quad_form.

## 3Description

Let $x$ have an $n$-dimensional multivariate Normal distribution with mean $\mu$ and variance-covariance matrix $\Sigma$. Then for a symmetric matrix $A$, g01nac computes up to the first $12$ moments and cumulants of the quadratic form $Q={x}^{\mathrm{T}}Ax$. The $s$th moment (about the origin) is defined as
 $E(Qs),$
where $E$ denotes expectation. The $s$th moment of $Q$ can also be found as the coefficient of ${t}^{s}/s!$ in the expansion of $E\left({e}^{Qt}\right)$. The $s$th cumulant is defined as the coefficient of ${t}^{s}/s!$ in the expansion of $\mathrm{log}\left(E\left({e}^{Qt}\right)\right)$.
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).
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

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{mom}$Nag_SelectMoments Input
On entry: indicates if moments are computed in addition to cumulants.
${\mathbf{mom}}=\mathrm{Nag_CumulantsOnly}$
Only cumulants are computed.
${\mathbf{mom}}=\mathrm{Nag_ComputeMoments}$
Moments are computed in addition to cumulants.
Constraint: ${\mathbf{mom}}=\mathrm{Nag_CumulantsOnly}$ or $\mathrm{Nag_ComputeMoments}$.
3: $\mathbf{mean}$Nag_IncludeMean Input
On entry: indicates if the mean, $\mu$, is zero.
${\mathbf{mean}}=\mathrm{Nag_MeanZero}$
$\mu$ is zero.
${\mathbf{mean}}=\mathrm{Nag_MeanInclude}$
The value of $\mu$ is supplied in emu.
Constraint: ${\mathbf{mean}}=\mathrm{Nag_MeanZero}$ or $\mathrm{Nag_MeanInclude}$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the dimension of the quadratic form.
Constraint: ${\mathbf{n}}>1$.
5: $\mathbf{a}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
the $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n×n$ symmetric matrix $A$. Only the lower triangle is referenced.
6: $\mathbf{pda}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
7: $\mathbf{emu}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array emu must be at least
• ${\mathbf{n}}$ when ${\mathbf{mean}}=\mathrm{Nag_MeanInclude}$;
• $1$ otherwise.
On entry: if ${\mathbf{mean}}=\mathrm{Nag_MeanInclude}$, emu must contain the $n$ elements of the vector $\mu$.
If ${\mathbf{mean}}=\mathrm{Nag_MeanZero}$, emu is not referenced.
8: $\mathbf{sigma}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array sigma must be at least ${\mathbf{pdsig}}×{\mathbf{n}}$.
the $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{sigma}}\left[\left(j-1\right)×{\mathbf{pdsig}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{sigma}}\left[\left(i-1\right)×{\mathbf{pdsig}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n×n$ variance-covariance matrix $\Sigma$. Only the lower triangle is referenced.
Constraint: the matrix $\Sigma$ must be positive definite.
9: $\mathbf{pdsig}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array sigma.
Constraint: ${\mathbf{pdsig}}\ge {\mathbf{n}}$.
10: $\mathbf{l}$Integer Input
On entry: the required number of cumulants, and moments if specified.
Constraint: $1\le {\mathbf{l}}\le 12$.
11: $\mathbf{rkum}\left[{\mathbf{l}}\right]$double Output
On exit: the l cumulants of the quadratic form.
12: $\mathbf{rmom}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array rmom must be at least
• ${\mathbf{l}}$ when ${\mathbf{mom}}=\mathrm{Nag_ComputeMoments}$;
• $1$ otherwise.
On exit: if ${\mathbf{mom}}=\mathrm{Nag_ComputeMoments}$, the l moments of the quadratic form.
13: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{l}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{l}}\le 12$.
On entry, ${\mathbf{l}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{l}}\ge 1$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>1$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdsig}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdsig}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdsig}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdsig}}\ge {\mathbf{n}}$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_POS_DEF
On entry, sigma is not positive definite.

## 7Accuracy

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

## 8Parallelism and Performance

g01nac makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

This example is given by Magnus and Pesaran (1993b) and considers the simple autoregression
 $yt=βyt-1+ut, t=1,2,…n,$
where $\left\{{u}_{t}\right\}$ is a sequence of independent Normal variables with mean zero and variance one, and ${y}_{0}$ is known. The moments of the quadratic form
 $Q=∑t=2nytyt-1$
are computed using g01nac. The matrix $A$ is given by:
 $A(i+1,i) = 12, i=1,2,…n-1; A(i,j) = 0, otherwise.$
The value of $\Sigma$ can be computed using the relationships
 $var(yt)=β2var(yt-1)+1$
and
 $cov(ytyt+k)=β cov(ytyt+k- 1)$
for $k\ge 0$ and $\mathrm{var}\left({y}_{1}\right)=1$.
The values of $\beta$, ${y}_{0}$, $n$, and the number of moments required are read in and the moments and cumulants printed.

### 10.1Program Text

Program Text (g01nace.c)

### 10.2Program Data

Program Data (g01nace.d)

### 10.3Program Results

Program Results (g01nace.r)