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High-Dimensional Mahalanobis Distances of Complex Random Vectors

Author

Listed:
  • Deliang Dai

    (Department of Economics and Statistics, Linnaeus University, 35195 Växjö, Sweden)

  • Yuli Liang

    (Department of Statistics, Örebro Univeristy, 70281 Örebro, Sweden)

Abstract

In this paper, we investigate the asymptotic distributions of two types of Mahalanobis distance (MD): leave-one-out MD and classical MD with both Gaussian- and non-Gaussian-distributed complex random vectors, when the sample size n and the dimension of variables p increase under a fixed ratio c = p / n → ∞ . We investigate the distributional properties of complex MD when the random samples are independent, but not necessarily identically distributed. Some results regarding the F-matrix F = S 2 − 1 S 1 —the product of a sample covariance matrix S 1 (from the independent variable array ( b e ( Z i ) 1 × n ) with the inverse of another covariance matrix S 2 (from the independent variable array ( Z j ≠ i ) p × n )—are used to develop the asymptotic distributions of MDs. We generalize the F-matrix results so that the independence between the two components S 1 and S 2 of the F-matrix is not required.

Suggested Citation

  • Deliang Dai & Yuli Liang, 2021. "High-Dimensional Mahalanobis Distances of Complex Random Vectors," Mathematics, MDPI, vol. 9(16), pages 1-12, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1877-:d:610095
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    References listed on IDEAS

    as
    1. D. Dai & T. Holgersson & P. Karlsson, 2017. "Expected and unexpected values of individual Mahalanobis distances," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(18), pages 8999-9006, September.
    2. Holgersson, Thomas & Dai, Deliang, 2014. "High-dimensional CLTs for individual Mahalanobis distances," Working Paper Series in Economics and Institutions of Innovation 361, Royal Institute of Technology, CESIS - Centre of Excellence for Science and Innovation Studies.
    3. Bai, Z. D. & Yin, Y. Q. & Krishnaiah, P. R., 1986. "On limiting spectral distribution of product of two random matrices when the underlying distribution is isotropic," Journal of Multivariate Analysis, Elsevier, vol. 19(1), pages 189-200, June.
    4. Birke, Melanie & Dette, Holger, 2005. "A note on testing the covariance matrix for large dimension," Statistics & Probability Letters, Elsevier, vol. 74(3), pages 281-289, October.
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