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

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.