Kernel Density Estimation and Convolution Revisited
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908606762319872 |
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| author | Tenkorang, Nicholas Ohene-Obeng, Kwesi Appau Su, Xiaogang |
| author_facet | Tenkorang, Nicholas Ohene-Obeng, Kwesi Appau Su, Xiaogang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19960 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kernel Density Estimation and Convolution Revisited Tenkorang, Nicholas Ohene-Obeng, Kwesi Appau Su, Xiaogang Methodology Statistics Theory 62G07, 62G20, 65D07 G.3 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. |
| title | Kernel Density Estimation and Convolution Revisited |
| topic | Methodology Statistics Theory 62G07, 62G20, 65D07 G.3 |
| url | https://arxiv.org/abs/2510.19960 |