naginterfaces.library.stat.moments_​quad_​form

naginterfaces.library.stat.moments_quad_form(a, sigma, l, emu=None)

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

For full information please refer to the NAG Library document for g01na

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g01/g01naf.html

Parameters

afloat, array-like, shape $(n,n)$

The $n\times n$ symmetric matrix $A$. Only the lower triangle is referenced.

sigmafloat, array-like, shape $(n,n)$

The $n\times n$ variance-covariance matrix $\Sigma$. Only the lower triangle is referenced.

lint

The required number of cumulants, and moments if specified.

emuNone or float, array-like, shape $(:)$, optional

Note: the required length for this argument is determined as follows: if $\text{mean}=\text{'M'}$: $n$; otherwise: $1$.

If $\text{mean}=\text{'M'}$, $emu$ must contain the $n$ elements of the vector $\mu$.

If $\text{mean}=\text{'Z'}$, $emu$ is not referenced.

Returns

rkumfloat, ndarray, shape $\left(l\right)$

The $l$ cumulants of the quadratic form.

rmomfloat, ndarray, shape $(:)$

If $\text{mom}=\text{'M'}$, the $l$ moments of the quadratic form.

Raises

NagValueError

(errno $1$)

On entry, $l=⟨value⟩$.

Constraint: $l\le 12$.

(errno $1$)

On entry, $\text{mean}=⟨value⟩$.

Constraint: $\text{mean}=\text{'Z'}$ or $\text{'M'}$.

(errno $1$)

On entry, $\text{mom}=⟨value⟩$.

Constraint: $\text{mom}=\text{'C'}$ or $\text{'M'}$.

(errno $1$)

On entry, $l=⟨value⟩$.

Constraint: $l\ge 1$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>1$.

(errno $2$)

On entry, $sigma$ is not positive definite.

Notes

Let $x$ have an $n$-dimensional multivariate Normal distribution with mean $\mu$ and variance-covariance matrix $\Sigma$. Then for a symmetric matrix $A$, moments_quad_form computes up to the first $12$ moments and cumulants of the quadratic form $Q=x^{T}Ax$. The $s$th moment (about the origin) is defined as

$$E\left(Q^s\right)\text{,}$$

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 $\log \left(E\left(e^{Qt}\right)\right)$.

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).

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, 1993, 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, 1993, The evaluation of moments of quadratic forms and ratios of quadratic forms in Normal variables: Background, motivation and examples, Comput. Statist. (8), 47–55