NAG Library Routine Document

1Purpose

g01naf computes the cumulants and moments of quadratic forms in Normal variates.

2Specification

Fortran Interface
 Subroutine g01naf ( mom, mean, n, a, lda, emu, l, rkum, rmom, wk,
 Integer, Intent (In) :: n, lda, ldsig, l Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a(lda,n), emu(*), sigma(ldsig,n) Real (Kind=nag_wp), Intent (Inout) :: rmom(*) Real (Kind=nag_wp), Intent (Out) :: rkum(l), wk(3*n*(n+1)/2+n) Character (1), Intent (In) :: mom, mean
#include <nagmk26.h>
 void g01naf_ (const char *mom, const char *mean, const Integer *n, const double a[], const Integer *lda, const double emu[], const double sigma[], const Integer *ldsig, const Integer *l, double rkum[], double rmom[], double wk[], Integer *ifail, const Charlen length_mom, const Charlen length_mean)

3Description

Let $x$ have an $n$-dimensional multivariate Normal distribution with mean $\mu$ and variance-covariance matrix $\Sigma$. Then for a symmetric matrix $A$, g01naf 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
 $EQs,$
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 routine is based on the routine CUM written by Magnus and Pesaran (1993a) and based on the theory given by Magnus (1978), Magnus (1979) and Magnus (1986).

4References

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:     $\mathbf{mom}$ – Character(1)Input
On entry: indicates if moments are computed in addition to cumulants.
${\mathbf{mom}}=\text{'C'}$
Only cumulants are computed.
${\mathbf{mom}}=\text{'M'}$
Moments are computed in addition to cumulants.
Constraint: ${\mathbf{mom}}=\text{'C'}$ or $\text{'M'}$.
2:     $\mathbf{mean}$ – Character(1)Input
On entry: indicates if the mean, $\mu$, is zero.
${\mathbf{mean}}=\text{'Z'}$
$\mu$ is zero.
${\mathbf{mean}}=\text{'M'}$
The value of $\mu$ is supplied in emu.
Constraint: ${\mathbf{mean}}=\text{'Z'}$ or $\text{'M'}$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the dimension of the quadratic form.
Constraint: ${\mathbf{n}}>1$.
4:     $\mathbf{a}\left({\mathbf{lda}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the $n$ by $n$ symmetric matrix $A$. Only the lower triangle is referenced.
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which g01naf is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
6:     $\mathbf{emu}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array emu must be at least ${\mathbf{n}}$ if ${\mathbf{mean}}=\text{'M'}$, and at least $1$ otherwise.
On entry: if ${\mathbf{mean}}=\text{'M'}$, emu must contain the $n$ elements of the vector $\mu$.
If ${\mathbf{mean}}=\text{'Z'}$, emu is not referenced.
7:     $\mathbf{sigma}\left({\mathbf{ldsig}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the $n$ by $n$ variance-covariance matrix $\Sigma$. Only the lower triangle is referenced.
Constraint: the matrix $\Sigma$ must be positive definite.
8:     $\mathbf{ldsig}$ – IntegerInput
On entry: the first dimension of the array sigma as declared in the (sub)program from which g01naf is called.
Constraint: ${\mathbf{ldsig}}\ge {\mathbf{n}}$.
9:     $\mathbf{l}$ – IntegerInput
On entry: the required number of cumulants, and moments if specified.
Constraint: $1\le {\mathbf{l}}\le 12$.
10:   $\mathbf{rkum}\left({\mathbf{l}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the l cumulants of the quadratic form.
11:   $\mathbf{rmom}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array rmom must be at least ${\mathbf{l}}$ if ${\mathbf{mom}}=\text{'M'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{mom}}=\text{'M'}$, the l moments of the quadratic form.
12:   $\mathbf{wk}\left(3×{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2+{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
13:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{l}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{l}}\ge 1$.
On entry, ${\mathbf{l}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{l}}\le 12$.
On entry, ${\mathbf{lda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{ldsig}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldsig}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{mean}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mean}}=\text{'Z'}$ or $\text{'M'}$.
On entry, ${\mathbf{mom}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mom}}=\text{'C'}$ or $\text{'M'}$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.
${\mathbf{ifail}}=2$
On entry, sigma is not positive definite.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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

g01naf 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 routine. 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 g01naf. The matrix $A$ is given by:
 $Ai+1,i = 12, i=1,2,…n-1; Ai,j = 0, otherwise.$
The value of $\Sigma$ can be computed using the relationships
 $varyt=β2varyt-1+1$
and
 $covytyt+k=β covytyt+k- 1$
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 (g01nafe.f90)

10.2Program Data

Program Data (g01nafe.d)

10.3Program Results

Program Results (g01nafe.r)