A Concentration of Measure and Random Matrix Approach to Large Dimensional Robust Statistics

The literature in this domain has so far divided the study of Ĉ into (i) a first exploration of conditions for its existence and uniqueness as a deterministic solution to (1) (e.g., [4, 8, 12]) and (ii) an independent analysis of its statistical properties when seen as a random object (in the large n regime [1] or in the large n, p regime [2, 14]). In the present article, we claim that the study of the conditions of existence (i) and statistical behavior (ii) of Ĉ can be conveniently carried out jointly. Specifically, by means of a flexible framework based on concentration of measure theory and on a new stable semimetric argument, we simultaneously explore the existence and large dimensional ( n, p large) spectral properties of Ĉ . Our findings may be summarized as the following three main contributions to robust statistics and more generally to large dimensional statistics.