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Entropy and density approximation from Laplace transforms

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  • Gzyl, Henryk
  • Novi Inverardi, Pierluigi
  • Tagliani, Aldo

Abstract

How much information does the Laplace transforms on the real line carry about an unknown, absolutely continuous distribution? If we measure that information by the Boltzmann–Gibbs–Shannon entropy, the original question becomes: How to determine the information in a probability density from the given values of its Laplace transform. We prove that a reliable evaluation both of the entropy and density can be done by exploiting some theoretical results about entropy convergence, that involve only finitely many real values of the Laplace transform, without having to invert the Laplace transform.We provide a bound for the approximation error of in terms of the Kullback–Leibler distance and a method for calculating the density to arbitrary accuracy.

Suggested Citation

  • Gzyl, Henryk & Novi Inverardi, Pierluigi & Tagliani, Aldo, 2015. "Entropy and density approximation from Laplace transforms," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 225-236.
    • Handle: RePEc:eee:apmaco:v:265:y:2015:i:c:p:225-236
    DOI: 10.1016/j.amc.2015.05.020
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    References listed on IDEAS

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    1. Mnatsakanov, Robert M., 2008. "Hausdorff moment problem: Reconstruction of distributions," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1612-1618, September.
    2. Henryk Gzyl & Pier Luigi Novi Inverardi & Aldo Tagliani, 2014. "Fractional Moments and Maximum Entropy: Geometric Meaning," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 43(17), pages 3596-3601, September.
    3. Mnatsakanov, Robert M., 2008. "Hausdorff moment problem: Reconstruction of probability density functions," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1869-1877, September.
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    Cited by:

    1. Dang, Chao & Xu, Jun, 2020. "Unified reliability assessment for problems with low- to high-dimensional random inputs using the Laplace transform and a mixture distribution," Reliability Engineering and System Safety, Elsevier, vol. 204(C).
    2. Zhang, Yang & Xu, Jun & Beer, Michael, 2023. "A single-loop time-variant reliability evaluation via a decoupling strategy and probability distribution reconstruction," Reliability Engineering and System Safety, Elsevier, vol. 232(C).

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