Robust test for detecting a signal in a high dimensional sparse normal vector

Let Zi, i=1,…,ni=1,…,n, be independent random variables, EZi=μiEZi=μi, and Var(Zi)=1Var(Zi)=1. We consider the problem of testing H0:μi=0,i=1,…,n, when n   is large, and the vector (μ1,…,μn)(μ1,…,μn) is ‘sparse’, e.g., ∑i=1nμi2=o(n). We suggest a robust test which is not sensitive to the exact tail behavior implied under normality assumptions. In particular, our test is ‘robust’ if the ‘moderate deviation’ tail of the distribution of Zi may be represented as the product of a tail of a standard normal and a ‘slowly changing’ function. This implies that whenever an Anderson–Darling type of test is robust our proposed test is also ‘robust’. A situation where the above mentioned tail behavior is expected, is when the Zi are of the form Zi=∑j=1mYij/m, for large m  , m⪡nm⪡n, and Yij are independent and identically distributed. For Zi of this form we show that our test is ‘robust’ when logn=o(m), while Anderson–Darling type tests are robust only when (logn)3=o(m). We provide examples and simulation evidence to demonstrate the robustness of our proposed test and the need for such robust tests. We also present a real data example highlighting the importance of robustness.