Exact minimax estimation of homogeneous complex Gaussian energies
- Primary subject
- Statistics and Data Science
- Subject subfields
- Statistical Theory · Statistical Inference
- Paper type
- Research Article
- Manuscript stage
- Preprint
Let f be a fixed nonzero homogeneous holomorphic polynomial of degree n on complex d-space. We estimate its Gaussian energy F_f(A) = E|f(X)|^2 from N independent centered proper complex Gaussian observations with unknown positive semidefinite covariance A. Singular covariances are included and F_f(A) > 0. Writing W = sum_s X_s X_s*, a = N(N+1)...(N+n-1) and b = N(N+1)...(N+2n-1), we prove that the exact minimax risk under relative squared error is 1-a^2/b, attained by the estimator (a/b)F_f(W). The unbiased estimator F_f(W)/a has exact worst-case relative variance b/a^2-1, with equality at rank-one covariances. A classical complex Wishart moment operator supplies the uniform upper bound, while a scalar scale subexperiment and proper inverse-Gamma priors give the matching lower bound against all estimators. Permanent estimation is a special case. Uniform relative-error estimation has observation complexity of order n^2 epsilon^(-2) log(1/delta), with computation unrestricted.