Edgeworth: Edgeworth expansion of a multivariate density

Description

Provides the truncated Edgeworth approximation to a multivariate density of $W = \sqrt{n} \bar{X}$ Approximation can use up to the first k=8 cumulants. The function implements the formula $$f_{\mathbf{W}^{\left( n\right) }}\left( \mathbf{w}\right) =\left( 1+\sum_{k=1}^{\infty }\frac{n^{-k/2}}{k!}\mathbf{B}_{k}\left( \frac{% \boldsymbol{\kappa }_{\mathbf{Y},3}^{\otimes \intercal }\mathbf{H}_{3}\left( \mathbf{z}|\mathbf{I}\right) }{6},\ldots ,\frac{ \boldsymbol{\kappa }_{\mathbf{Y},k+2}^{\otimes }\mathbf{H}_{k+2}\left( \mathbf{z}|\mathbf{I}\right) }{\left( k+1\right) \left( k+2\right) }\right) \right) \varphi \left( \mathbf{w}|\boldsymbol{\Sigma }_{\mathbf{X}}\right)$$ where $\mathbf{z}=\boldsymbol{\Sigma }_{\mathbf{X}}^{-1/2}(\mathbf{X}-\boldsymbol{\mu}_{\mathbf{X}})$, $\mathbf{B}_{k}$ denote the T-Bell Polynomials and $\varphi$ denotes the multivariate normal density. The case $n=1$ provides and approximation to the density of $\mathbf{X}$ and can be compared to the GramCharlier approximation.

Usage

Edgeworth(X, cum, n = 1)

Value

The vector of the Edgeworth 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.
n: the number of terms in the mean $\bar{\mathbf{X}}$

References

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

Examples

Run this code

# Edgeworth 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<-Edgeworth(X[1:50,],cum=EC,n=1)