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

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.