Simultaneous Inference in Non-Sparse High-Dimensional Linear Models

Inference and prediction under the sparsity assumption have been a hot research topic in recent years. However, in practice, the sparsity assumption is difficult to test, and more importantly can usually be violated. In this paper, to study hypothesis test of any group of parameters under non-sparse high-dimensional linear models, we transform the null hypothesis to a testable moment condition and then use the self-normalization structure to construct moment test statistics under one-sample and two-sample cases, respectively. Compared to the one-sample case, the two-sample additionally requires a convolution condition. It is worth noticing that these test statistics contain Modified Dantzig Selector, which simultaneously estimates model parameters and error variance without sparse assumption. Specifically, our method can be extended to heavy tailed distributions of error for its robustness. On very mild conditions, we show that the probability of Type I error is asymptotically equal to the nominal level {\alpha} and the probability of Type II error is asymptotically 0. Numerical experiments indicate that our proposed method has good finite-sample performance.