Breaking the curse of dimensionality in regression

The emergence of high-dimensional data, such as the gene expression values in microarray and the single nucleotide polymorphism data, brings challenges to many traditional statistical methods and theory. One important aspect of the high-dimensional data under the regression setting is that the number of covariates greatly exceeds the sample size. For example, in microarray data, the number of genes (p) is in the order of thousands whereas the sample size (n) is much less, usually less than 50. This is the so called "large-p, small-n" paradigm, which translates to a regime of asymptotics where p much faster than n. Inference in regression setting for large p, small n settings, have been recently developed. Sparsity assumption on the model signals has had a significant role in achieving optimal inference - Cai and Guo (2015); Javanmard and Montanari (2015); Cai and Guo (2016) found minimax results quantifying the direct effect of the size of the sparsity. In this article, we develop a test statistic that is able to quantify the simultaneous effect of a growing number of signals in a general high-dimensional linear model framework, allowing for a broad-ranging parameter structure.