🤖 AI Summary
This work addresses the challenges of bandwidth selection in kernel density estimation and its poor performance on multimodal, spiky, or discrete data by proposing a spectral method operating in the characteristic function domain. Treating bandwidth as a spectral truncation point, the approach integrates a data-driven noise-floor estimator, an adaptive Wiener tapering kernel, and a reconstruction architecture that combines a smooth basis with a band-limited residual. This framework unifies spectral decomposition, Wiener filtering, and Gaussian mixture modeling to enable automatic bandwidth selection and high-fidelity density estimation. Evaluated on the Marron–Wand benchmark suite (n = 5000), the method achieves the lowest integrated mean squared error and demonstrates superior robustness and reproducibility across six diverse real-world and synthetic datasets.
📝 Abstract
Kernel density estimation depends largely on one choice, the smoothing bandwidth. We treat bandwidth selection and density estimation in the characteristic-function domain, where the cyclic group-averaged covariance of the binned data has the squared empirical characteristic function as its spectrum: the true characteristic function sits over a sampling-noise floor of $1/n$, and the bandwidth is the spectral cutoff where the two meet. Several methods follow. An automatic selector strips the floor and minimizes a frequency-domain error criterion, matching the rule of thumb on smooth densities and approaching the best fixed bandwidth on multimodal ones. An adaptive estimator generalizes the fixed kernel to the per-frequency optimal Wiener taper, matching or surpassing the best fixed bandwidth on most standard densities, including sharply peaked and comb-like cases where fixed bandwidths fail; deconvolution under known measurement error follows in the same domain. Because the Wiener estimator resolves sharp structure but does not fit smooth bases as economically as a mixture, a Gaussian mixture is combined with it two ways, a piecewise partition and a superposition of a smooth base and a band-limited residual, the default. A data-driven floor read from the spectrum replaces the assumed $1/n$ floor and stays robust on heaped and rounded data. On the Marron-Wand benchmark scored by exact integrated squared error, the advantage emerges with sample size, a bias-variance tradeoff: the spectral estimators carry low bias but pay in variance, so cross-validation leads at $n=100$ while the Wiener filter and superposition take the top two ranks at $n=5000$. The methods are validated on six real datasets (CRSP returns, NHANES self-reports, CMS dimuon and SDSS spectra, a random-beacon stream, and UNSW-NB15 traffic) and on a synthetic-data quality check. All experiments are reproducible.