Moment-based Piecewise Polynomial Probability Density Estimation with Quantile-Based Binning

TL;DR

This paper introduces a quantile-based piecewise polynomial method for density estimation that improves accuracy and stability over traditional approaches, especially in tails, by combining local moments and equal-probability binning.

Contribution

It proposes a novel quantile-based piecewise polynomial density estimation technique with grid search for optimal binning and polynomial degrees, significantly reducing errors compared to existing methods.

Findings

Reduces K-S errors by 80-96% over standard polynomial methods.

Achieves comparable performance to kernel density estimation on real data.

Enhances control over tail behavior and oscillations in density estimates.

Abstract

Accurate reconstruction of probability density functions (PDFs) from data is essential in engineering applications. Classical global moment-based polynomial approximations often suffer from oscillations, instability in the tails, and sensitivity to the choice of support. This work proposes a quantile-based piecewise polynomial density reconstruction approach that combines equal-probability binning with local moment-matched polynomials within each bin. Two variants are considered: piecewise monomial and piecewise Lagrange polynomials with Chebyshev nodes. The numbers of bins and polynomial degrees are selected by a proposed grid search approach guided by the Kolmogorov-Smirnov (K-S) test statistic under non-negativity constraints. Across several benchmark distributions, the proposed methods reduce K-S errors by about $80$-$96\%$ relative to standard monomial and Lagrange polynomial approaches, and by about $83$-$97\%$ compared with spline density estimation. For real-world household electricity consumption and solar irradiance data, the piecewise approaches achieve K-S test statistic performance comparable to kernel density estimation while offering improved control over tail behavior and oscillations. Overall, the results demonstrate that quantile-based localization substantially enhances the robustness and fidelity of moment-based polynomial PDF reconstruction.