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Abstract:
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Let $Z_i, i=1,...,n$ independent random variables with mean $\mu_i$ and variance 1. We consider the problem of testing $H_0: \mu_i=0, i=1,...,n$. The setup is when $n$ is large, and the mean vector is `sparse.' We suggest a test which is not sensitive to the exact tail behavior implied under normality assumptions. In particular, if the 'moderate deviation' tail of the distribution of $Z_i$, may be represented as the product of a tail of a standard normal and a 'slowly changing' function, our suggested test is robust. Such a tail behavior, and a need for such a robust test, is expected when the $Z_i$ are of the form $Z_i=\sum_{j=1}^m Y_{ij}/ \sqrt{m}$, for large $m$, $m< < n$, and independent $Y_{ij}$.
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