Abstract
We study the signal detection problem in high dimensional noise data (possibly) containing rare and weak signals. Log-likelihood ratio (LLR) tests depend on unknown parameters, but they are needed to judge the quality of detection tests since they determine the detection regions. The popular Tukey’s higher criticism (HC) test was shown to achieve the same completely detectable region as the LLR test does for different (mainly) parametric models. We present a novel technique to prove this result for very general signal models, including even nonparametric $p$-value models. Moreover, we address the following questions which are still pending since the initial paper of Donoho and Jin: What happens on the border of the completely detectable region, the so-called detection boundary? Does HC keep its optimality there? In particular, we give a complete answer for the heteroscedastic normal mixture model. As a byproduct, we give some new insights about the LLR test’s behaviour on the detection boundary by discussing, among others, Pitmans’s asymptotic efficiency as an application of Le Cam’s theory.
Highlights
Signal detection in huge data sets becomes more and more important in current research
We study the signal detection problem in high dimensional noise data containing rare and weak signals
Log-likelihood ratio (LLR) tests depend on unknown parameters, but they are needed to judge the quality of detection tests since they determine the detection regions
Summary
Signal detection in huge data sets becomes more and more important in current research. The reason for HC’s popularity is that the area of complete detection coincide for the HC test and the log-likelihood ratio (LLR) test under different specific model assumption, see [2, 3, 5, 6, 11, 24]. There are (only) a few results concerning the asymptotic power behaviour of the LLR test on the detection boundary, which separates the area of complete detection and the area of no possible detection, see e.g. Its power behaviour is still optimal beyond the detection boundary for a long list of models.
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