F-distribution calibrated empirical likelihood ratio tests for multiple hypothesis testing

Multiple hypothesis testing can be important tools when conclusions are drawn by simultaneous testing of a large number of hypotheses in bioinformatics, general medicine, pharmacology and epidemiology. In this paper, we consider three nonparametric empirical likelihood ratio tests (ELRTs) for multiple hypothesis testing problems. When the number of hypotheses is far larger than sample size, however, these ELRTs using asymptotic chi-square calibration generally have much higher false discovery rate (FDR) and can be quite anti-conservative. We find that the first order term of the empirical likelihood ratio statistic closely resembles Hotelling's $T^2$T2 statistic admitting limiting F distributions for small sample size. Motivated by this result, we propose the F-distribution calibrated ELRTs. Simulation results indicate that the proposed tests not only can control the FDR in the acceptable range, but also guarantee the test efficacy in terms of maximising the number of discoveries for small and moderate sample sizes. Two real data applications are also included for illustration.