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Robust estimation methods for the mean vector, scatter matrix,and covariance matrix (if it exists) from data (possibly containing NAs)under multivariate heavy-tailed distributions such as angular Gaussian(via Tyler's method), Cauchy, and Student's t distributions. Additionally,a factor model structure can be specified for the covariance matrix. Thelatest revision also includes the multivariate skewed t distribution.

The package can be installed fromCRAN orGitHub:

# install stable version from CRANinstall.packages("fitHeavyTail")# install development version from GitHubdevtools::install_github("convexfi/fitHeavyTail")

To get help:

library(fitHeavyTail)help(package="fitHeavyTail")?fit_mvt

To citefitHeavyTail in publications:

citation("fitHeavyTail")

To illustrate the simple usage of the packagefitHeavyTail, let's start by generating some multivariate data under a Student's$t$ distribution with significant heavy tails (degrees of freedom$\nu=4$):

library(mvtnorm)# package for multivariate t distributionN<-10# number of variablesT<-80# number of observationsnu<-4# degrees of freedom for heavy tailsset.seed(42)mu<- rep(0,N)U<- t(rmvnorm(n= round(0.3*N),sigma=0.1*diag(N)))Sigma_cov<-U%*% t(U)+ diag(N)# covariance matrix with factor model structureSigma_scatter<- (nu-2)/nu*Sigma_covX<- rmvt(n=T,delta=mu,sigma=Sigma_scatter,df=nu)# generate data

We can first estimate the mean vector and covariance matrix via the traditional sample estimates (i.e., sample mean and sample covariance matrix):

mu_sm<- colMeans(X)Sigma_scm<- cov(X)

Then we can compute the robust estimates via the packagefitHeavyTail:

library(fitHeavyTail)fitted<- fit_mvt(X)

We can now compute the estimation errors and see the significant improvement:

sum((mu_sm-mu)^2)#> [1] 0.2857323sum((fitted$mu-mu)^2)#> [1] 0.1487845sum((Sigma_scm-Sigma_cov)^2)#> [1] 5.861138sum((fitted$cov-Sigma_cov)^2)#> [1] 4.663539

To get a visual idea of the robustness, we can plot the shapes of the covariance matrices (true and estimated ones) on two dimensions. Observe how the heavy-tailed estimation follows the true one more closely than the sample covariance matrix:

For more detailed information, please check thevignette.

Package:CRAN andGitHub.

README file:GitHub-readme.

Vignette:CRAN-vignette andGitHub-vignette.