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In big data analysis for detecting rare and weak signals among $n$ features,some grouping-test methods such as Higher Criticism test (HC), Berk-Jones test(B-J), and $\phi$-divergence test share the similar asymptotical optimalitywhen $n \rightarrow \infty$. However, in practical data analysis $n$ isfrequently small and moderately large at most. In order to properly apply theseoptimal tests and wisely choose them for practical studies, it is important toknow how to get the p-values and statistical power of them. To address thisproblem in an even broader context, this paper provides analytical solutionsfor a general family of goodness-of-fit (GOF) tests, which covers these optimaltests. For any given i.i.d. and continuous distributions of the input teststatistics of the $n$ features, both p-value and statistical power of such aGOF test can be calculated. By calculation we compared the finite-sampleperformances of asymptotically optimal tests under the normal mixturealternative. Results show that HC is the best choice when signals are rare,while B-J is more robust over various signal patterns. In the application to areal genome-wide association study, results illustrate that the p-valuecalculation works well, and the optimal tests have potentials for detectingnovel disease genes with weak genetic effects. The calculations have beenimplemented in an R package SetTest and published on the CRAN.

Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases