Research Article · Preprint Statistics and Data Science
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
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
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.
Keywords minimax estimation · Gaussian energy · complex Wishart · relative squared error · covariance functionals · permanent
Scope of the work
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.
Reproduction notes
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.
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Author disclosures
AI-assisted tools were used in mathematical exploration, drafting and checking. Responsibility for the statements and presentation rests with the author.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Exact minimax estimation of homogeneous complex Gaussian energies (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/exact-minimax-homogeneous-complex-gaussian-energies/v/1
BibTeX
@techreport{luminary_exact_minimax_homogeneous_complex_gaussian_energies_v1,
title = {Exact minimax estimation of homogeneous complex Gaussian energies},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/exact-minimax-homogeneous-complex-gaussian-energies/v/1}
}Loading discussion…