Kernel Density Estimation and Convolution Revisited

TL;DR

This paper introduces SHIDE, a new density estimation method that improves upon traditional KDE by using a convolutional framework, bounded noise, and spline interpolation, especially effective for bounded and heavy-tailed data.

Contribution

The paper presents SHIDE, a novel, computationally efficient density estimator with theoretical guarantees, extending KDE through a convolutional approach and boundary bias mitigation.

Findings

SHIDE attains the classical $n^{-4/5}$ convergence rate.

SHIDE reduces boundary bias compared to KDE.

Simulations show SHIDE performs well on bounded and heavy-tailed distributions.

Abstract

Kernel Density Estimation (KDE) is a cornerstone of nonparametric statistics, yet it remains sensitive to bandwidth choice, boundary bias, and computational inefficiency. This study revisits KDE through a principled convolutional framework, providing an intuitive model-based derivation that naturally extends to constrained domains, such as positive-valued random variables. Building on this perspective, we introduce SHIDE (Simulation and Histogram Interpolation for Density Estimation), a novel and computationally efficient density estimator that generates pseudo-data by adding bounded noise to observations and applies spline interpolation to the resulting histogram. The noise is sampled from a class of bounded polynomial kernel densities, constructed through convolutions of uniform distributions, with a natural bandwidth parameter defined by the kernel's support bound. We establish the theoretical properties of SHIDE, including pointwise consistency, bias-variance decomposition, and asymptotic MISE, showing that SHIDE attains the classical $n^{-4/5}$ convergence rate while mitigating boundary bias. Two data-driven bandwidth selection methods are developed, an AMISE-optimal rule and a percentile-based alternative, which are shown to be asymptotically equivalent. Extensive simulations demonstrate that SHIDE performs comparably to or surpasses KDE across a broad range of models, with particular advantages for bounded and heavy-tailed distributions. These results highlight SHIDE as a theoretically grounded and practically robust alternative to traditional KDE.