High-Dimensional Mahalanobis Distances of Complex Random Vectors

• Main Authors: Deliang Dai, Yuli Liang
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Mathematics
• Subjects: Mahalanobis distance; complex random vector; moments of MDs
• Online Access: https://www.mdpi.com/2227-7390/9/16/1877

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 <i>n</i> and the dimension of variables <i>p</i> increase under a fixed ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>=</mo><mi>p</mi><mo>/</mo><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">F</mi><mo>=</mo><msubsup><mi mathvariant="bold-italic">S</mi><mn>2</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi mathvariant="bold-italic">S</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>—the product of a sample covariance matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold-italic">S</mi><mn>1</mn></msub></semantics></math></inline-formula> (from the independent variable array <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="bold-italic">b</mi><mi>e</mi><msub><mrow><mo>(</mo><msub><mi>Z</mi><mi>i</mi></msub><mo>)</mo></mrow><mrow><mn>1</mn><mo>×</mo><mi>n</mi></mrow></msub></mrow></semantics></math></inline-formula>) with the inverse of another covariance matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold-italic">S</mi><mn>2</mn></msub></semantics></math></inline-formula> (from the independent variable array <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi mathvariant="bold-italic">Z</mi><mrow><mi>j</mi><mo>≠</mo><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>p</mi><mo>×</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula>)—are used to develop the asymptotic distributions of MDs. We generalize the F-matrix results so that the independence between the two components <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold-italic">S</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold-italic">S</mi><mn>2</mn></msub></semantics></math></inline-formula> of the F-matrix is not required.