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 = x^{T} A x

. The

s

th moment (about the origin) is defined as

E (Q^{s}),

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 (e^{Q t})

. The

s

th cumulant is defined as the coefficient of

t^{s} / s!

in the expansion of

\log (E (e^{Q t}))

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: $order$ – Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $mom$ – Nag_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

Nag_ComputeMoments

3: $mean$ – Nag_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

Nag_MeanInclude

4: $n$ – IntegerInput

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 \times n

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

n

symmetric matrix

A

. Only the lower triangle is referenced.

6: $pda$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq n

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

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 \times n

The

(i, j)

th element of the matrix is stored in

$sigma [(j - 1) \times pdsig + i - 1]$ when $order = Nag_ColMajor$ ;
$sigma [(i - 1) \times pdsig + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

n

variance-covariance matrix

Σ

. Only the lower triangle is referenced.

Constraint: the matrix

Σ

must be positive definite.

9: $pdsig$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array sigma.

Constraint:

pdsig \geq n

10: $l$ – IntegerInput

On entry: the required number of cumulants, and moments if specified.

Constraint:

1 \leq l \leq 12

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: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $l = 〈value〉$ .
Constraint: $l \leq 12$ .

On entry, $l = 〈value〉$ .
Constraint: $l \geq 1$ .

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: $pda \geq n$ .

On entry, $pdsig = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdsig \geq n$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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

Parallelism and Performance

nag_moments_quad_form (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.

9

Further Comments

None.

10

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_moments_quad_form (g01nac). 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.

NAG Library Function Document

nag_moments_quad_form (g01nac)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results