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Second Order Chebyshev–Edgeworth-Type Approximations for Statistics Based on Random Size Samples

Author

Listed:
  • Gerd Christoph

    (Department of Mathematics, Otto-von-Guericke University Magdeburg, 39016 Magdeburg, Germany
    These authors contributed equally to this work.)

  • Vladimir V. Ulyanov

    (Faculty of Computer Science, HSE University, 101000 Moscow, Russia
    Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    These authors contributed equally to this work.)

Abstract

This article completes our studies on the formal construction of asymptotic approximations for statistics based on a random number of observations. Second order Chebyshev–Edgeworth expansions of asymptotically normally or chi-squared distributed statistics from samples with negative binomial or Pareto-like distributed random sample sizes are obtained. The results can have applications for a wide spectrum of asymptotically normally or chi-square distributed statistics. Random, non-random, and mixed scaling factors for each of the studied statistics produce three different limit distributions. In addition to the expected normal or chi-squared distributions, Student’s t -, Laplace, Fisher, gamma, and weighted sums of generalized gamma distributions also occur.

Suggested Citation

  • Gerd Christoph & Vladimir V. Ulyanov, 2023. "Second Order Chebyshev–Edgeworth-Type Approximations for Statistics Based on Random Size Samples," Mathematics, MDPI, vol. 11(8), pages 1-18, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1848-:d:1122579
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    References listed on IDEAS

    as
    1. Gerd Christoph & Vladimir V. Ulyanov, 2021. "Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes," Mathematics, MDPI, vol. 9(7), pages 1-28, April.
    2. Christian Schluter & Mark Trede, 2016. "Weak convergence to the Student and Laplace distributions," Post-Print hal-01447853, HAL.
    3. Victor Korolev & Andrey Gorshenin, 2020. "Probability Models and Statistical Tests for Extreme Precipitation Based on Generalized Negative Binomial Distributions," Mathematics, MDPI, vol. 8(4), pages 1-30, April.
    4. Fujikoshi, Y. & Ulyanov, V.V. & Shimizu, R., 2005. "L1-norm error bounds for asymptotic expansions of multivariate scale mixtures and their applications to Hotelling's generalized," Journal of Multivariate Analysis, Elsevier, vol. 96(1), pages 1-19, September.
    5. Alexander Bulinski & Nikolay Slepov, 2022. "Sharp Estimates for Proximity of Geometric and Related Sums Distributions to Limit Laws," Mathematics, MDPI, vol. 10(24), pages 1-37, December.
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    Cited by:

    1. Alexander N. Tikhomirov & Vladimir V. Ulyanov, 2023. "On the Special Issue “Limit Theorems of Probability Theory”," Mathematics, MDPI, vol. 11(17), pages 1-4, August.

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