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Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes

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 Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    These authors contributed equally to this work.)

Abstract

Second-order Chebyshev–Edgeworth expansions are derived for various statistics from samples with random sample sizes, where the asymptotic laws are scale mixtures of the standard normal or chi-square distributions with scale mixing gamma or inverse exponential distributions. A formal construction of asymptotic expansions is developed. Therefore, the results can be applied to a whole family of asymptotically normal or chi-square statistics. The random mean, the normalized Student t -distribution and the Student t -statistic under non-normality with the normal limit law are considered. With the chi-square limit distribution, Hotelling’s generalized T 0 2 statistics and scale mixture of chi-square distributions are used. We present the first Chebyshev–Edgeworth expansions for asymptotically chi-square statistics based on samples with random sample sizes. The statistics allow non-random, random, and mixed normalization factors. Depending on the type of normalization, we can find three different limit distributions for each of the statistics considered. Limit laws are Student t -, standard normal, inverse Pareto, generalized gamma, Laplace and generalized Laplace as well as weighted sums of generalized gamma distributions. The paper continues the authors’ studies on the approximation of statistics for randomly sized samples.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:7:p:775-:d:529167
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    References listed on IDEAS

    as
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    4. Gerd Christoph & Vladimir V. Ulyanov, 2020. "Second Order Expansions for High-Dimension Low-Sample-Size Data Statistics in Random Setting," Mathematics, MDPI, vol. 8(7), pages 1-28, July.
    5. Safari, Muhammad Aslam Mohd & Masseran, Nurulkamal & Ibrahim, Kamarulzaman & Hussain, Saiful Izzuan, 2019. "A robust and efficient estimator for the tail index of inverse Pareto distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 517(C), pages 431-439.
    6. 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.
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    Cited by:

    1. 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.

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