GramCharlier: Gram-Charlier approximation to a multivariate density

Description

Provides the truncated Gram-Charlier approximation to a multivariate density. Approximation can be up to the first k=8 cumulants according to the formula $$f_{\mathbf{X}}\left( \mathbf{x}\right) =\left( 1+\sum_{k=3}^{\infty }\frac{1}{k!}\mathbf{B}_{k}^{\intercal }\left( 0,0, \boldsymbol{\kappa }_{\mathbf{Y},3}^{\otimes },\ldots \boldsymbol{\kappa }_{ \mathbf{Y},k}^{\otimes }\right) \mathbf{H}_{k}\left( \mathbf{y}|\mathbf{I} \right) \right) \varphi \left( \mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma }_{ \mathbf{X}}\right), $$ where the Hermite polynomial $\mathbf{H}_{k}\left( \mathbf{y}| \mathbf{I}\right) =\mathbf{H}_{k}\left( \boldsymbol{\Sigma }^{-1/2}\mathbf{x} \right) $ corresponds to the standard Gaussian variate, the cumulants are the cumulants of the standardized variate $\mathbf{Y}= \boldsymbol{\Sigma }^{-1/2}\left( \mathbf{X}-\boldsymbol{\mu }\right) $ of $\mathbf{X}$, ($\boldsymbol{\mu }={E}\mathbf{X}$) and $\varphi \left( \mathbf{x}\boldsymbol{|\boldsymbol{\mu},\Sigma }\right)$ denotes the multivariate normal density function with mean $\boldsymbol{\mu}$ and variance matrix $\boldsymbol{ \Sigma }$.

Usage

GramCharlier(X, cum)

Value

The vector of the Gram-Charlier density evaluated at X

Arguments

X: A matrix of d-variate data
cum: a list containing the raw (unstandardized) cumulant vectors of X. At least the first 3 cumulants need to be provided

References

Gy.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021. Section 4.7.

Examples

Run this code

# Gram-Charlier density approximation (k=4) of data generated from
# a bivariate skew-gaussian distribution
n<-500
alpha<-c(10,0)
omega<-diag(2)
X<-rSkewNorm(n,omega,alpha)
EC<-SampleMomCum(X,r=4,centering=FALSE,scaling=FALSE)
EC<-EC$estCum.r  ## (estimated) raw cumulants of X
fx4<-GramCharlier(X[1:50,],cum=EC)