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A0675
Title: Adaptive testing in high dimension Authors:  Xiaofeng Shao - University of Illinois at Urbana-Champaign (United States)
Yangfan Zhang - University of Illinois at Urbana-Champaign (United States)
Runmin Wang - Texas A&M University (United States) [presenting]
Abstract: A general $U$-statistic-based approach to adaptive testing for high-dimensional data is introduced. The proposed method extends a recent adaptive test by combining U-statistics for $l_q$ norm of the parameter vector with different $2\le q<\infty$, as the larger the $q$, the better the power against sparse alternatives. For a general parameter vector, it has been proved that the $U$-statistic for the $l_q$ norm is asymptotically normal under mild regularity conditions. More importantly, such $U$-statistics for different $q$ are still asymptotically independent, which has already been shown for the specific problems discussed previously. Further, a new test is developed only using subsamples with monotone indices to reduce the computational cost with mild efficiency loss. It was proved that the new method could speed up the calculation by a lot with mild efficiency loss. An application of the proposed test to change point detection will also be discussed. Simulation studies indicate that the new method is powerful against both dense and sparse alternatives for numerous problems.