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Depth based trimmed means

Alejandro Cholaquidis, Ricardo Fraiman, Leonardo Moreno, Gonzalo Perera

TL;DR

This work extends robust location estimation to high dimensions by formulating multivariate trimmed means through data depth functions. It proves almost sure consistency of the depth-based trimmed mean under mixing and establishes a general limit distribution for a broad family of depth-based estimators, which includes Tukey's and projection depth, without requiring Hadamard differentiability as an a priori assumption. The authors derive a Hadamard differentiability result for the trimmed-mean functional and provide the corresponding asymptotic distribution, expressed as a linear functional of the limiting density process. A simulation study illustrates the finite-sample behavior and the shape of the limiting distribution across different depth notions, highlighting practical guidance for applications in machine learning, economics, and financial risk assessment.

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.

Depth based trimmed means

TL;DR

This work extends robust location estimation to high dimensions by formulating multivariate trimmed means through data depth functions. It proves almost sure consistency of the depth-based trimmed mean under mixing and establishes a general limit distribution for a broad family of depth-based estimators, which includes Tukey's and projection depth, without requiring Hadamard differentiability as an a priori assumption. The authors derive a Hadamard differentiability result for the trimmed-mean functional and provide the corresponding asymptotic distribution, expressed as a linear functional of the limiting density process. A simulation study illustrates the finite-sample behavior and the shape of the limiting distribution across different depth notions, highlighting practical guidance for applications in machine learning, economics, and financial risk assessment.

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.
Paper Structure (9 sections, 9 theorems, 110 equations, 3 figures)

This paper contains 9 sections, 9 theorems, 110 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The multivariate trimmed mean
  3. Road map
  4. Main hypotheses
  5. On the level sets of $D$
  6. Consistency
  7. On the Haddamard differentiability of trimmed means.
  8. Asymptotic distribution for trimmed means
  9. Simulations

Key Result

Lemma 3

For each $i=1,\dots,\ell$, $\tau_i$ is a homeomorphism over its image, $\tau_i(\Omega_i)$, whose inverse is given by:

Figures (3)

  • Figure 1: Density estimation (left panel) and corresponding level sets (right panel) of the distribution of $\sqrt{a_n} \left( \Pi(D_n, f_n) - \Pi(D, f) \right)$ for a sample of size $500$, computed using Liu depth for different values of $a$.
  • Figure 2: Density estimation (left panel) and corresponding level sets (right panel) of the distribution of $\sqrt{a_n} \left( \Pi(D_n, f_n) - \Pi(D, f) \right)$ for a sample of size $500$, computed using Tukey depth for different values of $a$.
  • Figure 3: Density estimation (left panel) and corresponding level sets (right panel) of the distribution of $\sqrt{a_n} \left( \Pi(D_n, f_n) - \Pi(D, f) \right)$ for a sample of size $500$, computed using Projection depth for different values of $a$.

Theorems & Definitions (23)

  • Remark 1
  • Remark 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Remark 6
  • Theorem 7
  • ...and 13 more