Optimal Estimation of Co-heritability in High-dimensional Linear Models

Co-heritability is an important concept that characterizes the genetic associations within pairs of quantitative traits. There has been significant recent interest in estimating the co-heritability based on data from the genome-wide association studies (GWAS). This paper introduces two measures of co-heritability in the highdimensional linear model framework, including the inner product of the two regression vectors and a normalized inner product by their lengths. Functional de-biased estimators (FDEs) are developed to estimate these two co-heritability measures. In addition, estimators of quadratic functionals of the regression vectors are proposed. Both theoretical and numerical properties of the estimators are investigated. In particular, minimax rates of convergence are established and the proposed estimators of the inner product, the quadratic functionals and the normalized inner product are shown to be rate-optimal. Simulation results show that the FDEs significantly outperform the naive plug-in estimates. The FDEs are also applied to analyze a yeast segregant data set with multiple traits to estimate heritability and co-heritability among the traits. Zijian Guo is a Ph.D. student (Email: [email protected]). Wanjie Wang is a Postdoctoral Research Fellow (Email: [email protected]). T. Tony Cai is Dorothy Silberberg Professor of Statistics (E-mail: [email protected]). The research of T. Tony Cai was supported in part by NSF Grants DMS-1208982 and DMS-1403708, and NIH Grant R01 CA127334. Hongzhe Li is Professor (E-mail: [email protected]). The research of Wanjie Wang and Hongzhe Li was supported in part by NIH grants CA127334 and GM097505.