On some Diagonalized and Regularized Hotelling’s 𝑻𝟐 Tests of Location for High Dimensional Data

Abstract

A widely used statistical test of hypothesis for location parameter in ℝ𝑝is the Hotelling’s 𝑇2test. This test is efficient if data is normally distributed, ratio of sample size to dimension diverges and there are no outliers in the data. However, it is practically impossible to implementwhendimensionisgreaterthansamplesize.Asaremedialmeasure, diagonalized and regularized Hotelling’s 𝑇2tests were proposed. In this paper, powers of regularized and diagonalized Hotelling’s 𝑇2tests are compared with the usual Hotelling’s 𝑇2testin low dimension and the usual Hotelling’s 𝑇2perform much better. It is observed that diagonalized Hotelling’s 𝑇2test may have low power for mixture distributions. Due to a comparative performance of regularized and diagonalized Hotelling’s 𝑇2tests, robust versions of diagonalized and regularized Hotelling’s 𝑇2tests are proposed in high dimension in the presence of outliers. The powers of these tests were compared using simulated as well as real datasets.