Convergence and Fluctuations of Regularized Tyler Estimators

This paper studies the behavior of regularized Tyler estimators (RTEs) of scatter matrices. The advantages of these estimators are twofold. First, they guarantee by construction a good conditioning of the estimate and second, being derivatives of robust Tyler estimators, they inherit their robustness properties, notably their resilience to outliers. Nevertheless, one major problem that poses the use of RTEs is represented by the question of setting the regularization parameter
$\rho $
. While a high value of
$\rho $
is likely to push all the eigenvalues away from zero, it comes at the cost of a larger bias with respect to the population covariance matrix. A deep understanding of the statistics of RTEs is essential to come up with appropriate choices for the regularization parameter. This is not an easy task and requires working under asymptotic regimes wherein the number of observations
$n$
and/or their size
$N$
increase together. First asymptotic results have recently been obtained when
$N$
and
$n$
are large and commensurable. Interestingly, no results concerning the regime of
$n$
going to infinity with
$N$
fixed exist. This motivates our work. In particular, we prove in this paper that the RTEs converge to a deterministic matrix when
$n\to \infty $
with
$N$
fixed, which is expressed as a function of the theoretical covariance matrix. We also derive the fluctuations of the RTEs around this limit and establish that these fluctuations converge in distribution to a multivariate Gaussian distribution with parameters depending on the population covariance and the regularization coefficient.