# Australian & New Zealand Journal of Statistics

## Data‐adaptive test for high‐dimensional multivariate analysis of variance problem

### Summary

We focus on the high‐dimensional multivariate analysis of variance problem. Efficiencies of the existing methods for this problem depend highly on the alternative patterns: the ${L}_{2}$‐norm‐based methods are sensitive to dense alternatives and the ${L}_{\infty }$‐norm‐based methods are powerful against sparse alternatives. However, there is no method that is uniformly powerful under various alternative patterns. To overcome this deficiency, we propose an adaptive approach, which is powerful against different alternative patterns. First, we propose a family of tests that is based on the adjusted Lp‐norm, with different p. Combining the adjusted Lp‐norm‐based tests together, we build a data‐adaptive test statistic. The multiplier bootstrap is employed to approximate the limiting distribution of the test statistic and its validity is justified by theoretical analysis. A simulation study provides empirical evidence towards the conclusion that the data‐adaptive test performs well under both dense and sparse alternatives. The proposed test is then applied to a gene expression data set, associated with breast cancer, which illustrates its practical usefulness.

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