标题： 核密度估计与卷积再审视 显示英文标题

标题： Kernel Density Estimation and Convolution Revisited

摘要： 核密度估计（KDE）是非参数统计学的基石，但其仍然对带宽选择、边界偏差和计算效率敏感。 本研究通过一种系统的卷积框架重新审视KDE，提供了一个直观的基于模型的推导，该方法自然地扩展到约束域，例如正值随机变量。 基于这一观点，我们引入了SHIDE（用于密度估计的模拟与直方图插值），这是一种新颖且计算高效的密度估计器，通过向观测值添加有界噪声生成伪数据，并对生成的直方图应用样条插值。 噪声是从一类有界多项式核密度中采样的，这些核密度通过均匀分布的卷积构造而成，其自然的带宽参数由核的支持边界定义。 我们建立了SHIDE的理论性质，包括逐点一致性、偏差-方差分解和渐近MISE，表明SHIDE达到了经典的$n^{-4/5}$收敛速度，同时减轻了边界偏差。 开发了两种数据驱动的带宽选择方法，一种是AMISE最优规则，另一种是基于百分位数的替代方法，证明它们在渐近意义上是等价的。 大量模拟表明，SHIDE在广泛模型中的表现与KDE相当或更优，特别是在有界和重尾分布方面具有明显优势。 这些结果突显了SHIDE作为一种理论上有依据且实际稳健的传统KDE替代方法。 显示英文摘要

摘要： 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.