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Statistical Estimation of the Kullback–Leibler Divergence

Author

Listed:
  • Alexander Bulinski

    (Steklov Mathematical Institute of Russian Academy of Sciences, 119991 Moscow, Russia)

  • Denis Dimitrov

    (Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119234 Moscow, Russia)

Abstract

Asymptotic unbiasedness and L 2 -consistency are established, under mild conditions, for the estimates of the Kullback–Leibler divergence between two probability measures in R d , absolutely continuous with respect to (w.r.t.) the Lebesgue measure. These estimates are based on certain k -nearest neighbor statistics for pair of independent identically distributed (i.i.d.) due vector samples. The novelty of results is also in treating mixture models. In particular, they cover mixtures of nondegenerate Gaussian measures. The mentioned asymptotic properties of related estimators for the Shannon entropy and cross-entropy are strengthened. Some applications are indicated.

Suggested Citation

  • Alexander Bulinski & Denis Dimitrov, 2021. "Statistical Estimation of the Kullback–Leibler Divergence," Mathematics, MDPI, vol. 9(5), pages 1-36, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:544-:d:510564
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    References listed on IDEAS

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    1. Patricia Alonso Ruiz & Evgeny Spodarev, 2018. "Entropy-based Inhomogeneity Detection in Fiber Materials," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1223-1239, December.
    2. Li, Jinyang & Shang, Pengjian, 2018. "Time irreversibility of financial time series based on higher moments and multiscale Kullback–Leibler divergence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 502(C), pages 248-255.
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    Cited by:

    1. Cadirci, Mehmet Siddik & Evans, Dafydd & Leonenko, Nikolai & Makogin, Vitalii, 2022. "Entropy-based test for generalised Gaussian distributions," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    2. 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.

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