Optimal Estimation of A Quadratic Functional and Detection of Simultaneous Signals
Tony Cai and Xin Lu Tan
Motivated by applications in genomics, this paper studies the problem of optimal estimation of a quadratic functional of two normal mean vectors, Q(μ, θ) =(1/n) Σni=1 μi2 θi2, with a particular focus on the case where both mean vectors are sparse. We propose optimal estimators of Q(μ, θ) for different regimes and establish the minimax rates of convergence over a family of parameter spaces. The optimal rates exhibit interesting phase transitions in this family. The problem of simultaneous signal detection is also considered under the minimax framework. It is shown that the proposed estimators for Q(μ, θ) naturally lead to optimal testing procedures.