Asymptotic expansion of the minimum covariance determinant estimators

TL;DR

This paper establishes the asymptotic behavior of minimum covariance determinant estimators for multivariate distributions with densities, providing explicit derivatives, symmetry conditions, and proving asymptotic normality with a detailed covariance structure.

Contribution

It proves the existence and explicit form of the derivative needed for asymptotic expansion of MCD estimators under broad conditions, including symmetry and elliptically contoured models.

Findings

Derivatives exist for distributions with densities.

Asymptotic normality of MCD estimators is established.

Explicit covariance structure for the estimators is derived.

Abstract

In arXiv:0907.0079 by Cator and Lopuhaa, an asymptotic expansion for the MCD estimators is established in a very general framework. This expansion requires the existence and non-singularity of the derivative in a first-order Taylor expansion. In this paper, we prove the existence of this derivative for multivariate distributions that have a density and provide an explicit expression. Moreover, under suitable symmetry conditions on the density, we show that this derivative is non-singular. These symmetry conditions include the elliptically contoured multivariate location-scatter model, in which case we show that the minimum covariance determinant (MCD) estimators of multivariate location and covariance are asymptotically equivalent to a sum of independent identically distributed vector and matrix valued random elements, respectively. This provides a proof of asymptotic normality and a precise description of the limiting covariance structure for the MCD estimators.