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.

翻译：对于高度偏斜或厚尾分布，基于均值或中位数的方法通常无法捕捉数据的中心趋势。尽管条件模式估计（给定某些协变量）是一种可行的替代方案，但模式回归面临重大挑战。非参数方法受制于"维度灾难"，而半参数策略往往导致非凸优化问题。为避免这些问题，我们提出了一种新颖的模式回归估计器，其依赖于条件分位数密度求逆的中间步骤。与现有方法不同，我们采用了一种卷积型平滑变体的分位数回归。我们的估计器在协变量设计点上具有一致收敛性，且与以往基于分位数的模式回归不同，其收敛性关于平滑带宽具有一致性。此外，卷积模式回归具有维度无关性，不存在优化问题，初步模拟实验表明该估计器在有限样本中服从正态分布。