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The Calculation of the Density and Distribution Functions of Strictly Stable Laws

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  • Viacheslav Saenko

    (Department of Theoretical Physics, Ulyanovsk State University, 432970 Ulyanovsk, Russia)

Abstract

Integral representations for the probability density and distribution function of a strictly stable law with the characteristic function in the Zolotarev’s “C” parametrization were obtained in the paper. The obtained integral representations express the probability density and distribution function of standard strictly stable laws through a definite integral. Using the methods of numerical integration, the obtained integral representations allow us to calculate the probability density and distribution function of a strictly stable law for a wide range of admissible values of parameters ( α , θ ) . A number of cases were given when numerical algorithms had difficulty in calculating the density. Formulas were given to calculate the density and distribution function with an arbitrary value of the scale parameter λ .

Suggested Citation

  • Viacheslav Saenko, 2020. "The Calculation of the Density and Distribution Functions of Strictly Stable Laws," Mathematics, MDPI, vol. 8(5), pages 1-38, May.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:5:p:775-:d:356869
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    References listed on IDEAS

    as
    1. Nolan, John P., 1998. "Parameterizations and modes of stable distributions," Statistics & Probability Letters, Elsevier, vol. 38(2), pages 187-195, June.
    2. Royuela-del-Val, Javier & Simmross-Wattenberg, Federico & Alberola-López, Carlos, 2017. "libstable: Fast, Parallel, and High-Precision Computation of α-Stable Distributions in R, C/C++, and MATLAB," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 78(i01).
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    Cited by:

    1. Viacheslav V. Saenko, 2022. "The Calculation of the Probability Density and Distribution Function of a Strictly Stable Law in the Vicinity of Zero," Mathematics, MDPI, vol. 10(20), pages 1-27, October.

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