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Polynomial probability distribution estimation using the method of moments

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  • Joakim Munkhammar
  • Lars Mattsson
  • Jesper Rydén

Abstract

We suggest a procedure for estimating Nth degree polynomial approximations to unknown (or known) probability density functions (PDFs) based on N statistical moments from each distribution. The procedure is based on the method of moments and is setup algorithmically to aid applicability and to ensure rigor in use. In order to show applicability, polynomial PDF approximations are obtained for the distribution families Normal, Log-Normal, Weibull as well as for a bimodal Weibull distribution and a data set of anonymized household electricity use. The results are compared with results for traditional PDF series expansion methods of Gram–Charlier type. It is concluded that this procedure is a comparatively simple procedure that could be used when traditional distribution families are not applicable or when polynomial expansions of probability distributions might be considered useful approximations. In particular this approach is practical for calculating convolutions of distributions, since such operations become integrals of polynomial expressions. Finally, in order to show an advanced applicability of the method, it is shown to be useful for approximating solutions to the Smoluchowski equation.

Suggested Citation

  • Joakim Munkhammar & Lars Mattsson & Jesper Rydén, 2017. "Polynomial probability distribution estimation using the method of moments," PLOS ONE, Public Library of Science, vol. 12(4), pages 1-14, April.
    • Handle: RePEc:plo:pone00:0174573
    DOI: 10.1371/journal.pone.0174573
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    References listed on IDEAS

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    1. Munkhammar, Joakim & Rydén, Jesper & Widén, Joakim, 2014. "Characterizing probability density distributions for household electricity load profiles from high-resolution electricity use data," Applied Energy, Elsevier, vol. 135(C), pages 382-390.
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    Cited by:

    1. Ignacio Algredo-Badillo & José Julio Conde-Mones & Carlos Arturo Hernández-Gracidas & María Monserrat Morín-Castillo & José Jacobo Oliveros-Oliveros & Claudia Feregrino-Uribe, 2020. "An FPGA-based analysis of trade-offs in the presence of ill-conditioning and different precision levels in computations," PLOS ONE, Public Library of Science, vol. 15(6), pages 1-26, June.
    2. Jacobi, Arie & Tzur, Joseph, 2021. "Wealth Distribution across Countries: Quality of Weibull, Dagum and Burr XII in Estimating Wealth over Time," Finance Research Letters, Elsevier, vol. 43(C).
    3. Munir Ali Elfarra & Mustafa Kaya, 2018. "Comparison of Optimum Spline-Based Probability Density Functions to Parametric Distributions for the Wind Speed Data in Terms of Annual Energy Production," Energies, MDPI, vol. 11(11), pages 1-15, November.
    4. Gong, Yu & Liu, Pan & Ming, Bo & Feng, Maoyuan & Huang, Kangdi & Wang, Yibo, 2022. "Identifying the functional form of operating rules for hydro–photovoltaic hybrid power systems," Energy, Elsevier, vol. 243(C).
    5. Jenny Farmer & Donald Jacobs, 2018. "High throughput nonparametric probability density estimation," PLOS ONE, Public Library of Science, vol. 13(5), pages 1-29, May.
    6. Yue Yu & Pavel Loskot, 2023. "Polynomial Distributions and Transformations," Mathematics, MDPI, vol. 11(4), pages 1-28, February.
    7. Michel Denuit & Christian Y. Robert, 2022. "Polynomial Series Expansions and Moment Approximations for Conditional Mean Risk Sharing of Insurance Losses," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 693-711, June.

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