Abstract: This paper studies the problem of optimal estimation of a quadratic functional of two normal mean vectors, , with a particular focus on the case where both mean vectors are sparse. We propose optimal estimators of 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. We also include a simulation study to complement the theoretical results in the paper.

Key words and phrases: Gaussian sequence model, minimax estimation, quadratic functional, signal detection, sparse means.