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Second Order Expansions for High-Dimension Low-Sample-Size Data Statistics in Random Setting

Author

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  • Gerd Christoph

    (Department of Mathematics, Otto-von-Guericke University Magdeburg, 39016 Magdeburg, Germany
    Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia
    These authors contributed equally to this work.)

  • Vladimir V. Ulyanov

    (Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Faculty of Computer Science, National Research University Higher School of Economics, 167005 Moscow, Russia
    These authors contributed equally to this work.)

Abstract

We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error bounds. Depending on the type of normalization, we get three different limit distributions: Normal, Student’s t -, or Laplace distributions. The paper continues studies of the authors on approximation of statistics for random size samples.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1151-:d:384168
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    References listed on IDEAS

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    3. Goldfeld, Stephen M. & Quandt, Richard E., 1981. "Econometric modelling with non-normal disturbances," Journal of Econometrics, Elsevier, vol. 17(2), pages 141-155, November.
    4. Christian Schluter & Mark Trede, 2016. "Weak convergence to the Student and Laplace distributions," Post-Print hal-01447853, HAL.
    5. H. M. Barakat & E. M. Nigm & Magdy E. El-Adll & M. Yusuf, 2018. "Prediction of future generalized order statistics based on exponential distribution with random sample size," Statistical Papers, Springer, vol. 59(2), pages 605-631, June.
    6. Konishi, Sadanori, 1979. "Asymptotic expansions for the distributions of functions of a correlation matrix," Journal of Multivariate Analysis, Elsevier, vol. 9(2), pages 259-266, June.
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    Cited by:

    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.

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