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Integrating the summary statistics from genome-wide association study(\textsc{gwas}) and expression quantitative trait loci (e\textsc{qtl}) dataprovides a powerful way of identifying the genes whose expression levels arepotentially associated with complex diseases. A parameter called $T$-score thatquantifies the genetic overlap between a gene and the disease phenotype basedon the summary statistics is introduced based on the mean values of twoGaussian 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-smoothfunctional, which characterizes the amount of shared signals between twoabsolute normal mean vectors $|\theta|$ and $|\mu|$. Using approximationtheory, estimators are constructed and shown to be minimax rate-optimal andadaptive over various parameter spaces. Simulation studies demonstrate thesuperiority of the proposed estimators over existing methods. The method isapplied to an integrative analysis of heart failure genomics datasets and weidentify several genes and biological pathways that are potentially causal tohuman heart failure.

Optimal Estimation of Simultaneous Signals Using Absolute Inner Product with Applications to Integrative Genomics