NAG Library Function Document

nag_moments_ratio_quad_forms (g01nbc)

nag_moments_ratio_quad_forms (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)

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

, nag_moments_ratio_quad_forms (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 a

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 nag_1d_quad_inf_1 (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_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

Nag_ColMajor

2: ratio_type – Nag_MomentTypeInput

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_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: b[ $\dim$ ] – const doubleInput

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 – IntegerInput

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 doubleInput

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

$\max (1, pdc \times n)$ when $ratio_type = Nag_QuadRatio$ ;
$1$ otherwise.

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 – IntegerInput

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

Constraints:

if $ratio_type = Nag_QuadRatio$ , $pdc \geq n$ ;
otherwise $pdc \geq 1$ .

11: ela[ $\dim$ ] – const doubleInput

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 a is not referenced.

12: 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.

13: 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.

14: 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

15: l1 – IntegerInput

On entry: the first moment to be computed,

l_{1}

Constraint:

0 < l1 \leq l2

16: l2 – IntegerInput

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[ $l2 - l1 + 1$ ] – doubleOutput

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 – doubleInput

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ACCURACY: Full accuracy not achieved in integration.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_EIGENVALUES: Failure in computing eigenvalues.
NE_ENUM_INT: On entry, $ratio_type = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $n > 0$ .
NE_ENUM_INT_2: On entry, $ratio_type = ⟨value⟩$ , $pdc = ⟨value⟩$ , $n = ⟨value⟩$ .
Constraint: if $ratio_type = Nag_QuadRatio$ , $pdc \geq n$ ;
otherwise $pdc \geq 1$ .
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.
NE_MOMENTS: Only $⟨value⟩$ moments exist, less than $l1 = ⟨value⟩$ .
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 $⟨value⟩$ moments exist, less than $l2 = ⟨value⟩$ .

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

nag_moments_ratio_quad_forms (g01nbc) is not threaded by NAG in any implementation.

nag_moments_ratio_quad_forms (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 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 nag_moments_ratio_quad_forms (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.

NAG Library Function Documentnag_moments_ratio_quad_forms (g01nbc)

+− 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 Library Function Document

nag_moments_ratio_quad_forms (g01nbc)