  High dimensional hypothesis test deals with models in which the number of
parameters is significantly larger than the sample size. Existing literature
develops a variety of individual tests. Some of them are sensitive to the dense
and small disturbance, and others are sensitive to the sparse and large
disturbance. Hence, the powers of these tests depend on the assumption of the
alternative scenario. This paper provides a unified framework for developing
new tests which are adaptive to a large variety of alternative scenarios in
high dimensions. In particular, our framework includes arbitrary hypotheses
which can be tested using high dimensional $U$-statistic based vectors. Under
this framework, we first develop a broad family of tests based on a novel
variant of the $L_p$-norm with $p\in \{1,\dots,\infty\}$. We then combine these
tests to construct a data-adaptive test that is simultaneously powerful under
various alternative scenarios. To obtain the asymptotic distributions of these
tests, we utilize the multiplier bootstrap for $U$-statistics. In addition, we
consider the computational aspect of the bootstrap method and propose a novel
low-cost scheme. We prove the optimality of the proposed tests. Thorough
numerical results on simulated and real datasets are provided to support our
theory.
