  Integrating the summary statistics from genome-wide association study
(\textsc{gwas}) and expression quantitative trait loci (e\textsc{qtl}) data
provides a powerful way of identifying the genes whose expression levels are
potentially associated with complex diseases. A parameter called $T$-score that
quantifies the genetic overlap between a gene and the disease phenotype based
on the summary statistics is introduced based on the mean values of two
Gaussian sequences. Specifically, given two independent samples
$\mathbf{x}_n\sim N(\theta, \Sigma_1)$ and $\mathbf{y}_n\sim N(\mu, \Sigma_2)$,
the $T$-score is defined as $\sum_{i=1}^n |\theta_i\mu_i|$, a non-smooth
functional, which characterizes the amount of shared signals between two
absolute normal mean vectors $|\theta|$ and $|\mu|$. Using approximation
theory, estimators are constructed and shown to be minimax rate-optimal and
adaptive over various parameter spaces. Simulation studies demonstrate the
superiority of the proposed estimators over existing methods. The method is
applied to an integrative analysis of heart failure genomics datasets and we
identify several genes and biological pathways that are potentially causal to
human heart failure.
