Central limit theorem and influence function for the MCD estimators at general multivariate distributions

TL;DR

This paper establishes the theoretical properties of the minimum covariance determinant estimators, including their existence, continuity, asymptotic normality, and influence functions, for general multivariate distributions.

Contribution

It provides a comprehensive theoretical analysis of MCD estimators, including new proofs of existence, continuity, and asymptotic normality in a broad multivariate context.

Findings

Existence of MCD functionals at any multivariate distribution

Continuity and separating ellipsoid property of the estimators

Asymptotic normality and influence functions derived

Abstract

We define the minimum covariance determinant functionals for multivariate location and scatter through trimming functions and establish their existence at any multivariate distribution. We provide a precise characterization including a separating ellipsoid property and prove that the functionals are continuous. Moreover, we establish asymptotic normality for both the location and covariance estimator and derive the influence function. These results are obtained in a very general multivariate setting.