Root approach for estimation of statistical distributions

Application of root density estimator to problems of statistical data analysis is demonstrated. Four sets of basis functions based on Chebyshev-Hermite, Laguerre, Kravchuk and Charlier polynomials are considered. The sets may be used for numerical analysis in problems of reconstructing statistical distributions by experimental data. Based on the root approach to reconstruction of statistical distributions and quantum states, we study a family of statistical distributions in which the probability density is the product of a Gaussian distribution and an even-degree polynomial. Examples of numerical modeling are given. The results of present paper are of interest for the development of tomography of quantum states and processes.

💡 Research Summary

The paper introduces a novel statistical tool called the root density estimator and demonstrates its effectiveness for reconstructing probability distributions and quantum states from experimental data. Traditional density‑estimation techniques such as kernel methods or histograms often suffer from over‑smoothing, negative probability values, and poor performance on discrete or high‑dimensional data. By contrast, the root approach writes a probability density (p(x)) as the squared modulus of a complex‑valued function (\psi(x)):
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