  In this article, we propose a class of $L_q$-norm based U-statistics for a
family of global testing problems related to high-dimensional data. This
includes testing of mean vector and its spatial sign, simultaneous testing of
linear model coefficients, and testing of component-wise independence for
high-dimensional observations, among others. Under the null hypothesis, we
derive asymptotic normality and independence between $L_q$-norm based
U-statistics for several $q$s under mild moment and cumulant conditions. A
simple combination of two studentized $L_q$-based test statistics via their
$p$-values is proposed and is shown to attain great power against alternatives
of different sparsity. Our work is a substantial extension of He et al. (2021),
which is mostly focused on mean and covariance testing, and we manage to
provide a general treatment of asymptotic independence of $L_q$-norm based
U-statistics for a wide class of kernels. To alleviate the computation burden,
we introduce a variant of the proposed U-statistics by using the monotone
indices in the summation, resulting in a U-statistic with asymmetric kernel. A
dynamic programming method is introduced to reduce the computational cost from
$O(n^{qr})$, which is required for the calculation of the full U-statistic, to
$O(n^r)$ where $r$ is the order of the kernel. Numerical studies further
corroborate the advantage of the proposed adaptive test as compared to some
existing competitors.
