Statistica Sinica 27 (2017), 879-906
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.