Exact minimax estimation of homogeneous complex Gaussian energies
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Abstract
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.
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Research details
- Subjects
- Statistics and Data Science · Statistical Theory · Statistical Inference
- Keywords
- minimax estimation, Gaussian energy, complex Wishart, relative squared error, covariance functionals, permanent
- Manuscript date
- 2026-10-26
- Reader reuse
- No additional permission granted
Scope
Exact finite-sample minimax relative squared risk for homogeneous holomorphic energies under unknown Hermitian positive semidefinite covariance, including singular covariances; permanent estimation and uniform confidence observation order as applications.
Disclosures
AI-assisted tools were used in mathematical exploration, drafting and checking. Responsibility for the statements and presentation rests with the author.
Reproduction
The source package includes the complete editable manuscript and a standard-library Python checker with 21 exact finite moment and shrinkage-risk diagnostics. Build and execution instructions are in README.md.