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Asymptotic Properties of a Statistical Estimator of the Jeffreys Divergence: The Case of Discrete Distributions

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  • Vladimir Glinskiy

    (Department of Business Analytics, Siberian Institute of Management—Branch of the Russian Presidential Academy of National Economy and Public Administration, Novosibirsk State University of Economics and Management, 630102 Novosibirsk, Russia
    Department of Statistics, Novosibirsk State University of Economics and Management, 630099 Novosibirsk, Russia)

  • Artem Logachov

    (Department of Business Analytics, Siberian Institute of Management—Branch of the Russian Presidential Academy of National Economy and Public Administration, Novosibirsk State University of Economics and Management, 630102 Novosibirsk, Russia
    Department of Computer Science in Economics, Novosibirsk State Technical University (NSTU), 630087 Novosibirsk, Russia)

  • Olga Logachova

    (Department of Higher Mathematics, Siberian State University of Geosystems and Technologies (SSUGT), 630108 Novosibirsk, Russia)

  • Helder Rojas

    (Escuela Profesional de Ingeniería Estadística, Universidad Nacional de Ingeniería, Lima 00051, Peru
    Department of Mathematics, Imperial College London, London SW7 2AZ, UK)

  • Lyudmila Serga

    (Department of Business Analytics, Siberian Institute of Management—Branch of the Russian Presidential Academy of National Economy and Public Administration, Novosibirsk State University of Economics and Management, 630102 Novosibirsk, Russia
    Department of Statistics, Novosibirsk State University of Economics and Management, 630099 Novosibirsk, Russia)

  • Anatoly Yambartsev

    (Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo (USP), São Paulo 05508-220, Brazil)

Abstract

We investigate the asymptotic properties of the plug-in estimator for the Jeffreys divergence, the symmetric variant of the Kullback–Leibler (KL) divergence. This study focuses specifically on the divergence between discrete distributions. Traditionally, estimators rely on two independent samples corresponding to two distinct conditions. However, we propose a one-sample estimator where the condition results from a random event. We establish the estimator’s asymptotic unbiasedness (law of large numbers) and asymptotic normality (central limit theorem). Although the results are expected, the proofs require additional technical work due to the randomness of the conditions.

Suggested Citation

  • Vladimir Glinskiy & Artem Logachov & Olga Logachova & Helder Rojas & Lyudmila Serga & Anatoly Yambartsev, 2024. "Asymptotic Properties of a Statistical Estimator of the Jeffreys Divergence: The Case of Discrete Distributions," Mathematics, MDPI, vol. 12(21), pages 1-16, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:21:p:3319-:d:1504748
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    References listed on IDEAS

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    1. David M. Blei & Alp Kucukelbir & Jon D. McAuliffe, 2017. "Variational Inference: A Review for Statisticians," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 859-877, April.
    2. Vexler, Albert & Gurevich, Gregory, 2010. "Empirical likelihood ratios applied to goodness-of-fit tests based on sample entropy," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 531-545, February.
    3. Alexander Bulinski & Denis Dimitrov, 2021. "Statistical Estimation of the Kullback–Leibler Divergence," Mathematics, MDPI, vol. 9(5), pages 1-36, March.
    4. Helder Rojas & Artem Logachov & Anatoly Yambartsev, 2023. "Order Book Dynamics with Liquidity Fluctuations: Asymptotic Analysis of Highly Competitive Regime," Mathematics, MDPI, vol. 11(20), pages 1-24, October.
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