Sparse-limit approximation for t-statistics

In a range of genomic applications, it is of interest to quantify the evidence that the signal at site~$i$ is active given conditionally independent replicate observations summarized by the sample mean and variance $(\bar Y, s^2)$ at each site. We study the version of the problem in which the signal distribution is sparse, and the error distribution has an unknown site-specific variance so that the null distribution of the standardized statistic is Student-$t$ rather than Gaussian. The main contribution of this paper is a sparse-mixture approximation to the non-null density of the $t$-ratio. This formula demonstrates the effect of low degrees of freedom on the Bayes factor, or the conditional probability that the site is active. We illustrate some differences on a HIV dataset for gene-expression data previously analyzed by Efron (2012).