Abstract
To date, only frequentist, Bayesian and empirical Bayes approaches have been studied for the large-scale inference problem of testing simultaneously hundreds or thousands of hypotheses. Their derivations start with some summarizing statistics without modelling the basic responses. As a consequence testing procedures have been developed without necessarily checking model assumptions, and empirical null distributions are needed to avoid the problem of rejecting all null hypotheses when the sample sizes are large. Nevertheless these procedures may not be statistically efficient. We present the multiple-testing problem as a multiple-prediction problem of whether a null hypothesis is true or not. We introduce hierarchical random-effect models for basic responses and show how the extended likelihood is built. It is shown that the likelihood prediction has a certain oracle property. The extended likelihood leads to new testing procedures, which are optimal for the usual loss function in hypothesis testing. The new tests are based on certain shrinkage t-statistics and control the local probability of false discovery for individual tests to maintain the global frequentist false discovery rate and have no need to consider an empirical null distribution for the shrinkage t-statistics. Conditions are given when these false rates vanish. Three examples illustrate how to use the likelihood method in practice. A numerical study shows that the likelihood approach can greatly improve existing methods and finding the best fitting model is crucial for the behaviour of test procedures.