Asymptotic Properties of the Misclassification Errors for Euclidean Distance Discriminant Rule in High-Dimensional Data

TL;DR

This paper investigates the asymptotic behavior of the misclassification error of the Euclidean Distance Discriminant Rule in high-dimensional settings where the dimension exceeds sample size, providing new theoretical insights and practical methods.

Contribution

It introduces new asymptotic distributions and unbiased estimators for misclassification errors of EDDR in high dimensions, along with methods to determine optimal cut-off points.

Findings

Asymptotic normality of quadratic forms established

Explicit formulas for misclassification error estimators derived

Numerical results confirm high accuracy in large, high-dimensional samples

Abstract

Performance accuracy of the Euclidean Distance Discriminant rule (EDDR) is studied in the high-dimensional asymptotic framework which allows the dimensionality to exceed sample size. Under mild assumptions on the traces of the covariance matrix, our new results provide the asymptotic distribution of the conditional misclassification error and the explicit expression for the consistent and asymptotically unbiased estimator of the expected misclassification error. To get these properties, new results on the asymptotic normality of the quadratic forms and traces of the higher power of Wishart matrix, are established. Using our asymptotic results, we further develop two generic methods of determining a cut-off point for EDDR to adjust the misclassification errors. Finally, we numerically justify the high accuracy of our asymptotic findings along with the cut-off determination methods in finite sample applications, inclusive of the large sample and high-dimensional scenarios.