The “large $p$, small $n$” paradigm arises in microarray studies, image analysis, high throughput molecular screening, astronomy, and in many other high dimensional applications. False discovery rate (FDR) methods are useful for resolving the accompanying multiple testing problems. In cDNA microarray studies, for example, $p$-values may be computed for each of $p$ genes using data from $n$ arrays, where typically $p$ is in the thousands and $n$ is less than 30. For FDR methods to be valid in identifying differentially expressed genes, the $p$-values for the nondifferentially expressed genes must simultaneously have uniform distributions marginally. While feasible for permutation $p$-values, this uniformity is problematic for asymptotic based $p$-values since the number of $p$-values involved goes to infinity and intuition suggests that at least some of the $p$-values should behave erratically. We examine this neglected issue when $n$ is moderately large but $p$ is almost exponentially large relative to $n$. We show the somewhat surprising result that, under very general dependence structures and for both mean and median tests, the $p$-values are simultaneously valid. A small simulation study and data analysis are used for illustration.
"Marginal asymptotics for the “large $p$, small $n$” paradigm: With applications to microarray data." Ann. Statist. 35 (4) 1456 - 1486, August 2007. https://doi.org/10.1214/009053606000001433