Convolution Mode Regression

TL;DR

This paper introduces a novel convolution-based mode regression estimator that effectively estimates the conditional mode in skewed or fat-tailed distributions, overcoming common challenges of existing methods.

Contribution

It proposes a convolution-type smoothed quantile regression approach that is dimension-free, convex, and uniformly convergent over covariates and bandwidths.

Findings

Estimator converges uniformly over covariates

Method is dimension-free and avoids non-convex optimization

Simulations show the estimator is normally distributed in finite samples

Abstract

For highly skewed or fat-tailed distributions, mean or median-based methods often fail to capture the central tendencies in the data. Despite being a viable alternative, estimating the conditional mode given certain covariates (or mode regression) presents significant challenges. Nonparametric approaches suffer from the "curse of dimensionality", while semiparametric strategies often lead to non-convex optimization problems. In order to avoid these issues, we propose a novel mode regression estimator that relies on an intermediate step of inverting the conditional quantile density. In contrast to existing approaches, we employ a convolution-type smoothed variant of the quantile regression. Our estimator converges uniformly over the design points of the covariates and, unlike previous quantile-based mode regressions, is uniform with respect to the smoothing bandwidth. Additionally, the Convolution Mode Regression is dimension-free, carries no issues regarding optimization and preliminary simulations suggest the estimator is normally distributed in finite samples.