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Abstract

Let f be a fixed nonzero homogeneous holomorphic polynomial of degree n on ℂ^d. 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=Σ_{s=1}^N X_sX_s^*, a=(N rising n) and b=(N rising2n), 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^2ε^(-2)log(1/δ), with computation unrestricted.