Sparsity Implications for Multiple Testing

Music City Center 

Consider a set of observations from a common exponential family governed by potentially different univariate parameters. We wish to test separate null and alternative hypotheses for each parameter; the null hypothesis is that the parameter takes the value zero, and the alternative is that the value exceeds a common threshold. In most applications the threshold is called the signal strength. Suppose we know a limit on the number of alternatives that can be true. Common losses are based on the number of mistakes and often involve False Discovery Rate and False Non-discovery Rate. Most results tend to show asymptotic optimality of procedures. Here we investigate the impact of knowledge of maximal number of true alternatives for fixed sample size.