g01nbc (Nag_OrderType order, Nag_MomentType ratio_type, Nag_IncludeMean mean, Integer n, const double a[], Integer pda, const double b[], Integer pdb, const double c[], Integer pdc, const double ela[], const double emu[], const double sigma[], Integer pdsig, Integer l1, Integer l2, Integer *lmax, double rmom[], double *abserr, double eps, NagError *fail)

The function may be called by the names: g01nbc, nag_stat_moments_ratio_quad_forms or nag_moments_ratio_quad_forms.

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

and symmetric positive semidefinite matrix

B

, g01nbc computes a subset,

l_{1}

l_{2}

, of the first

12

moments of the ratio of quadratic forms

R = x^{T} A x / x^{T} B x .

The

s

th moment (about the origin) is defined as

E (R^{s}),

(1)

where

E

denotes the expectation. Alternatively, this function will compute the following expectations:

E (R^{s} (a^{T} x))

(2)

and

E (R^{s} (x^{T} C x)),

(3)

where

a

is a vector of length

n

and

C

is an

n

n

symmetric matrix, if they exist. In the case of (2) the moments are zero if

μ = 0

The conditions of theorems 1, 2 and 3 of Magnus (1986) and Magnus (1990) are used to check for the existence of the moments. If all the requested moments do not exist, the computations are carried out for those moments that are requested up to the maximum that exist,

l_{MAX}

This function is based on the function QRMOM written by Magnus and Pesaran (1993a) and based on the theory given by Magnus (1986) and Magnus (1990). The computation of the moments requires first the computation of the eigenvectors of the matrix

L^{T} B L

, where

L L^{T} = Σ

. The matrix

L^{T} B L

must be positive semidefinite and not null. Given the eigenvectors of this matrix, a function which has to be integrated over the range zero to infinity can be computed. This integration is performed using d01smc.

4 References

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 (1990) On certain moments relating to quadratic forms in Normal variables: Further results Sankhyā, Ser. B 52 1–13

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_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

order = 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:

order = Nag_RowMajor

Nag_ColMajor

2: $ratio_type$ – Nag_MomentType Input

On entry: indicates the moments of which function are to be computed.

$ratio_type = Nag_RatioMoments$ (Ratio): $E (R^{s})$ is computed.
$ratio_type = Nag_LinearRatio$ (Linear with ratio): $E (R^{s} (a^{T} x))$ is computed.
$ratio_type = Nag_QuadRatio$ (Quadratic with ratio): $E (R^{s} (x^{T} C x))$ is computed.

Constraint:

ratio_type = Nag_RatioMoments

Nag_LinearRatio

Nag_QuadRatio

3: $mean$ – Nag_IncludeMean Input

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: $n$ – Integer Input

On entry:

n

, the dimension of the quadratic form.

Constraint:

n > 1

5: $a [\dim]$ – const double Input

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: $pda$ – Integer Input

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

Constraint:

pda \geq n

7: $b [\dim]$ – const double Input

Note: the dimension, dim, of the array b must be at least

pdb \times n

the

(i, j)

th element of the matrix

B

is stored in

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

On entry: the

n

n

positive semidefinite symmetric matrix

B

. Only the lower triangle is referenced.

Constraint: the matrix

B

must be positive semidefinite.

8: $pdb$ – Integer Input

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

Constraint:

pdb \geq n

9: $c [\dim]$ – const double Input

Note: the dimension, dim, of the array c must be at least

n

when

ratio_type = Nag_QuadRatio

The

(i, j)

th element of the matrix

C

is stored in

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

On entry: if

ratio_type = Nag_QuadRatio

, c must contain the

n

n

symmetric matrix

C

; only the lower triangle is referenced.

ratio_type \neq Nag_QuadRatio

, c is not referenced.

10: $pdc$ – Integer Input

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

Constraint: if

ratio_type = Nag_QuadRatio

pdc \geq n

11: $ela [\dim]$ – const double Input

Note: the dimension, dim, of the array ela must be at least

$n$ when $ratio_type = Nag_LinearRatio$ ;
$1$ otherwise.

On entry: if

ratio_type = Nag_LinearRatio

, ela must contain the vector

a

of length

n

, otherwise ela is not referenced.

12: $emu [\dim]$ – const double Input

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.

13: $sigma [\dim]$ – const double Input

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.

14: $pdsig$ – Integer Input

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

Constraint:

pdsig \geq n

15: $l1$ – Integer Input

On entry: the first moment to be computed,

l_{1}

Constraint:

0 < l1 \leq l2

16: $l2$ – Integer Input

On entry: the last moment to be computed,

l_{2}

Constraint:

l1 \leq l2 \leq 12

17: $lmax$ – Integer * Output

On exit: the highest moment computed,

l_{MAX}

. This will be

l_{2}

on successful exit.

18: $rmom [l2 - l1 + 1]$ – double Output

On exit: the

l_{1}

l_{MAX}

moments.

19: $abserr$ – double * Output

On exit: the estimated maximum absolute error in any computed moment.

20: $eps$ – double Input

On entry: the relative accuracy required for the moments, this value is also used in the checks for the existence of the moments.

eps = 0.0

, a value of

\sqrt{ε}

where

ε

is the machine precision used.

Constraint:

eps = 0.0

eps \geq machine precision

21: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ACCURACY: The required accuracy has not been achieved in the integration. An estimate of the accuracy is returned in abserr.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_EIGENVALUES: The computation to compute the eigenvalues required in the calculation of moments has failed to converge: this is an unlikely error exit.
NE_ENUM_INT: On entry, $ratio_type = 〈value〉$ and $n = 〈value〉$ .
Constraint: $n > 0$ .
NE_ENUM_INT_2: On entry, $ratio_type = 〈value〉$ , $pdc = 〈value〉$ and $n = 〈value〉$ .
Constraint:
if $ratio_type = Nag_QuadRatio$ , $pdc \geq n$ .
NE_INT: On entry, $l1 = 〈value〉$ .
Constraint: $l1 \geq 1$ .

On entry, $l2 = 〈value〉$ .
Constraint: $l2 \leq 12$ .

On entry, $n = 〈value〉$ .
Constraint: $n > 1$ .

On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .

On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .

On entry, $pdc = 〈value〉$ .
Constraint: $pdc > 0$ .

On entry, $pdsig = 〈value〉$ .
Constraint: $pdsig > 0$ .
NE_INT_2: On entry, $l1 = 〈value〉$ and $l2 = 〈value〉$ .
Constraint: $0 < l1 \leq l2$ .

On entry, $l1 = 〈value〉$ and $l2 = 〈value〉$ .
Constraint: $l1 \leq l2 \leq 12$ .

On entry, $l1 = 〈value〉$ and $l2 = 〈value〉$ .
Constraint: $l2 \geq l1$ .

On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq n$ .

On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq n$ .

On entry, $pdc = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdc \geq n$ .

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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MOMENTS: Only $〈value〉$ moments exist, less than $l1 = 〈value〉$ , therefore none of the required moments can be computed.
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.
NE_POS_SEMI_DEF: On entry, b is not positive semidefinite or is null.

The matrix $L^{T} B L$ is not positive semidefinite or is null.
NE_REAL: On entry, $eps = 〈value〉$ .
Constraint: if $eps \neq 0.0$ , $eps \geq machine precision$ .
NE_SOME_MOMENTS: Only some of the required moments have been computed, the highest is given by lmax.

7 Accuracy

The relative accuracy is specified by eps and an estimate of the maximum absolute error for all computed moments is returned in abserr.

8 Parallelism and Performance

g01nbc 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 least squares estimate of

β

\hat{β}

, is given by

\hat{β} = \frac{\sum_{t = 2}^{n} y_{t} y_{t - 1}}{\sum_{t = 2}^{n} y_{t}^{2}} .

Thus

\hat{β}

can be written as a ratio of quadratic forms and its moments computed using g01nbc. The matrix

A

is given by

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

and the matrix

B

is given by

\begin{matrix} B (i, i) = 1, & i = 1, 2, \dots n - 1; \\ B (i, j) = 0, & otherwise. \end{matrix}

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 computed and printed.

Interfaces: FL CL AD

NAG CL Interface Introduction

G01 (Stat) Chapter Contents

G01 (Stat) Chapter Introduction

g01nb: FL CL AD

NAG CL Interfaceg01nbc (moments_​ratio_​quad_​forms)

▸▿ 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

NAG CL Interface
g01nbc (moments_ratio_quad_forms)