# The Minimum Regularized Covariance Determinant estimator

## TL;DR

The paper introduces the MRCD estimator, an extension of MCD that is applicable in high-dimensional settings, providing robust, well-conditioned covariance estimates with a fast algorithm for outlier detection.

## Contribution

It proposes the MRCD method, combining regularization with MCD, applicable in any dimension, and develops a fast algorithm with proven reduction steps.

## Key findings

- MRCD is effective in high-dimensional outlier detection.
- The estimator maintains robustness and good conditioning.
- Simulation and real data demonstrate practical utility.

## Abstract

The Minimum Covariance Determinant (MCD) approach robustly estimates the location and scatter matrix using the subset of given size with lowest sample covariance determinant. Its main drawback is that it cannot be applied when the dimension exceeds the subset size. We propose the Minimum Regularized Covariance Determinant (MRCD) approach, which differs from the MCD in that the scatter matrix is a convex combination of a target matrix and the sample covariance matrix of the subset. A data-driven procedure sets the weight of the target matrix, so that the regularization is only used when needed. The MRCD estimator is defined in any dimension, is well-conditioned by construction and preserves the good robustness properties of the MCD. We prove that so-called concentration steps can be performed to reduce the MRCD objective function, and we exploit this fact to construct a fast algorithm. We verify the accuracy and robustness of the MRCD estimator in a simulation study and illustrate its practical use for outlier detection and regression analysis on real-life high-dimensional data sets in chemistry and criminology.