Depth based trimmed means

TL;DR

This paper extends robust univariate trimmed mean techniques to multivariate data using depth functions, proving consistency and limit distributions for these estimators, with practical implications across various fields.

Contribution

It introduces theoretical foundations for multivariate depth-based trimmed means, including consistency and limit distribution theorems, advancing robust high-dimensional location estimation.

Findings

Proved almost sure consistency of multivariate trimmed mean estimators.

Established a general limit distribution theorem for depth-based estimators.

Demonstrated practical application with simulated data varying depth measures and trimming levels.

Abstract

Robust estimation of location is a fundamental problem in statistics, particularly in scenarios where data contamination by outliers or model misspecification is a concern. In univariate settings, methods such as the sample median and trimmed means balance robustness and efficiency by mitigating the influence of extreme observations. This paper extends these robust techniques to the multivariate context through the use of data depth functions, which provide a natural means to order and rank multidimensional data. We review several depth measures and discuss their role in generalizing trimmed mean estimators beyond one dimension. Our main contributions are twofold: first, we prove the almost sure consistency of the multivariate trimmed mean estimator under mixing conditions; second, we establish a general limit distribution theorem for a broad family of depth-based estimators, encompassing popular examples such as Tukey's and projection depth. These theoretical advancements not only enhance the understanding of robust location estimation in high-dimensional settings but also offer practical guidelines for applications in areas such as machine learning, economic analysis, and financial risk assessment. A small example with simulated data is performed, varying the depth measure used and the percentage of trimmed data.