High-Dimensional Mahalanobis Distances of Complex Random Vectors

Deliang Dai and Yuli Liang

Mathematics, 2021, vol. 9, issue 16, 1-12

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.

Keywords: Mahalanobis distance; complex random vector; moments of MDs (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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