Summary. Estimation of the number or proportion of true null hypotheses in multiple-testing problems has become an interesting area of research. The first important work in this field was performed by Schweder and Spjøtvoll. Among others, they proposed to use plug-in estimates for the proportion of true null hypotheses in multiple-test procedures to improve the power. We investigate the problem of controlling the familywise error rate FWER when such estimators are used as plug-in estimators in single-step or step-down multiple-test procedures. First we investigate the case of independent p-values under the null hypotheses and show that a suitable choice of plug-in estimates leads to control of FWER in single-step procedures. We also investigate the power and study the asymptotic behaviour of the number of false rejections. Although step-down procedures are more difficult to handle we briefly consider a possible solution to this problem. Anyhow, plug-in step-down procedures are not recommended here. For dependent p-values we derive a condition for asymptotic control of FWER and provide some simulations with respect to FWER and power for various models and hypotheses.
Instead of defining goodness of fit (GOF) tests in terms of their test statistics, we present an alternative method by introducing the concept of local levels, which indicate high or low local sensitivity of a test. Local levels can act as a starting point for the construction of new GOF tests. We study the behavior of local levels when applied to some well-known GOF tests such as Kolmogorov-Smirnov (KS) tests, higher criticism (HC) tests and tests based on phi-divergences. The main focus is on a rigorous characterization of the asymptotic behavior of local levels of the original HC tests which leads to several further asymptotic results for local levels of other GOF tests including GOF tests with equal local levels. While local levels of KS tests, which are related to the central range, are asymptotically strictly larger than zero, all local levels of HC tests converge to zero as the sample size increases. Consequently, there exists no asymptotic level α GOF test such that all local levels are asymptotically bounded away from zero. Finally, by means of numerical computations we compare classical KS and HC tests to a GOF test with equal local levels.
The higher criticism (HC) statistic, which can be seen as a normalized version of the famous Kolmogorov-Smirnov statistic, has a long history, dating back to the mid seventies. Originally, HC statistics were used in connection with goodness of fit (GOF) tests but they recently gained some attention in the context of testing the global null hypothesis in high dimensional data. The continuing interest for HC seems to be inspired by a series of nice asymptotic properties related to this statistic. For example, unlike Kolmogorov-Smirnov tests, GOF tests based on the HC statistic are known to be asymptotically sensitive in the moderate tails, hence it is favorably applied for detecting the presence of signals in sparse mixture models. However, some questions around the asymptotic behavior of the HC statistic are still open. We focus on two of them, namely, why a specific intermediate range is crucial for GOF tests based on the HC statistic and why the convergence of the HC distribution to the limiting one is extremely slow. Moreover, the inconsistency in the asymptotic and finite behavior of the HC statistic prompts us to provide a new HC test that has better finite properties than the original HC test while showing the same asymptotics. This test is motivated by the asymptotic behavior of the so-called local levels related to the original HC test. By means of numerical calculations and simulations we show that the new HC test is typically more powerful than the original HC test in normal mixture models.
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