Abstract
We consider the problem of detecting sparse heterogeneous mixtures from a nonparametric perspective. Specifically, we assume that the null distribution is symmetric about zero, while the true effects have positive median. We then suggest two new tests for this purpose. The main one is a form of Anderson–Darling test for symmetry and is closely related to the higher criticism. It is shown to achieve the detection boundary for the normal mixture model and, more generally, for asymptotically generalized Gaussian mixture models, in all sparsity regimes. The other test is a form of longest run test and specifically designed for the very sparse situation.
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Notes
This literature typically assumes that \(F=G\) but this is not essential. Indeed, the definition of the HC test does not require knowledge of G and it was shown in (Cai and Wu 2014) that the HC can adapt to various Gs.
This was suggested to us by a reviewer.
The power under the mixture model (2) is a weighed average of the power at each m, where the weights are given by the binomial distribution with parameters \((n, \varepsilon )\).
In our theoretical developments, we focused on the risk, which is standard in the literature. In our numerical experiments, we chose instead to fix the level and evaluate the power. We did so because this is typically what is done in practice. In fact, optimizing the risk necessitates knowledge of the alternative, which is rarely available.
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Acknowledgments
We would like to thank Jason Schweinsberg for helpful discussions. This work was partially supported by a Grant from the US Office of Naval Research (N00014-09-1-0258) and a Grant from the US National Science Foundation (DMS 1223137).
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Arias-Castro, E., Wang, M. Distribution-free tests for sparse heterogeneous mixtures. TEST 26, 71–94 (2017). https://doi.org/10.1007/s11749-016-0499-x
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DOI: https://doi.org/10.1007/s11749-016-0499-x
Keywords
- Mixture detection
- Distribution-free tests
- Higher criticism
- Anderson–Darling test
- Smirnov test for symmetry