Mode-based estimation of the center of symmetry

TL;DR

This paper introduces a new nonparametric mode-based estimator for the center of symmetry, demonstrating its robustness and efficiency through theoretical analysis and empirical validation, outperforming traditional methods in various settings.

Contribution

It develops a novel mode-based estimator with a new kernel family, bridging nonparametrics and robust statistics, and provides theoretical and empirical evidence of its advantages.

Findings

The estimator outperforms the sample mean in heavy-tailed distributions.

A new one-parameter kernel family with compact support enhances robustness.

Empirical results show superior performance compared to traditional methods.

Abstract

In the mean-median-mode triad of univariate centrality measures, the mode has been overlooked for estimating the center of symmetry in continuous and unimodal settings. This paper expands on the connection between kernel mode estimators and M-estimators for location, bridging the gap between the nonparametrics and robust statistics communities. The variance of modal estimators is studied in terms of a bandwidth parameter, establishing conditions for an optimal solution that outperforms the household sample mean. A purely nonparametric approach is adopted, modeling heavy-tailedness through regular variation. The results lead to an estimator proposal that includes a novel one-parameter family of kernels with compact support, offering extra robustness and efficiency. The effectiveness and versatility of the new method are demonstrated in a real-world case study and a thorough simulation study, comparing favorably to traditional and more competitive alternatives. Several myths about the mode are clarified along the way, reopening the quest for flexible and efficient nonparametric estimators.