Mean Vector and Covariance Matrix Estimation under Heavy Tails

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This vignette illustrates the usage of the package fitHeavyTail to estimate the mean vector and covariance matrix of heavy-tailed multivariate distributions such as the angular Gaussian, Cauchy, or Student’s \(t\) distribution. The results are compared against existing benchmark functions from different packages.

Installation

The package can be installed from CRAN or GitHub:

# install stable version from CRAN
install.packages("fitHeavyTail")

# install development version from GitHub
devtools::install_github("convexfi/fitHeavyTail")

To get help:

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

To cite fitHeavyTail in publications:

citation("fitHeavyTail")

Quick Start

To illustrate the simple usage of the package fitHeavyTail, 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 distribution
N <- 10   # number of variables
T <- 80   # number of observations
nu <- 4   # degrees of freedom for heavy tails

set.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 structure
Sigma_scatter <- (nu-2)/nu * Sigma_cov
X <- 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 package fitHeavyTail:

library(fitHeavyTail)
fitted <- fit_mvt(X)

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

sum((mu_sm     - mu)^2)
#> [1] 0.2857323
sum((fitted$mu - mu)^2)
#> [1] 0.1487845

sum((Sigma_scm  - Sigma_cov)^2)
#> [1] 5.861138
sum((fitted$cov - Sigma_cov)^2)
#> [1] 3.031499

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:

Numerical Comparison with Existing Packages

In the following, we generate multivariate heavy-tailed Student’s \(t\) distributed data and compare the performance of many different existing packages via 100 Monte Carlo simulations in terms of estimation accurary, measured by the mean squared error (MSE) and CPU time.

The following plot gives a nice overall perspective of the MSE vs. CPU time tradeoff of the different methods (note the ellipse at the bottom left that embraces the best four methods: fitHeavyTail::fit_Tyler(), fitHeavyTail::fit_Cauchy(), fitHeavyTail::fit_mvt(), and fitHeavyTail::fit_mvt() with fixed nu = 6):