Randomized empirical likelihood test for ultra-high dimensional means under general covariances

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Abstract This article proposes a calibrated empirical likelihood test for ultra-high dimensional means that incorporates multiple projections. Under weak moment conditions on the distributions of data, we analyse all possible asymptotic distributions of the proposed test statistic in different scenarios. To determine the critical value and enhance test power, we employ the random symmetrization method based on the group of sign flips and use multiple selected projections. The test can still maintain the significance level asymptotically, even in the presence of heterogeneity in the data distribution. Moreover, the proposed test procedure allows for general covariance structures and ultra-high dimensional regimes. Further, the power function reveals the relation with the projection term in an asymptotic sense such that we can select suitable projections to achieve good power in various scenarios. A quasi-Newton algorithm is introduced to reduce the computational cost arising from the intensive optimizations required for computing empirical likelihood. Numerical studies evidence the promising performance of the proposed test compared with existing tests.