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Research Article · Preprint Statistics and Data Science

Exact minimax estimation of homogeneous complex Gaussian energies

Scott Schubr

Luminary Oddities

  • Statistics and Data Science
  • Statistical Theory
  • Statistical Inference
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
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Overview

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.

Keywords minimax estimation · Gaussian energy · complex Wishart · relative squared error · covariance functionals · permanent

Scope of the work

For each fixed nonzero degree-n homogeneous holomorphic energy of a proper complex Gaussian with unknown PSD covariance and positive energy, exact minimax relative squared risk is 1−(N rising n)²/(N rising 2n), including singular covariances.

Reproduction notes

The 8-page manuscript, editable source and citation metadata are supplied in this 6-member package. README.md gives build and execution commands. The Python standard-library checker Gaussian moment controls.py supplies the exact finite controls specified in the manuscript alongside its general analytic proofs.

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AI-assisted tools were used in mathematical exploration, drafting and checking. Responsibility for the statements and presentation rests with the author.

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Cite this edition

Scott Schubr. (2026). Exact minimax estimation of homogeneous complex Gaussian energies (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/exact-minimax-homogeneous-complex-gaussian-energies/v/2

BibTeX
@techreport{luminary_exact_minimax_homogeneous_complex_gaussian_energies_v2,
  title = {Exact minimax estimation of homogeneous complex Gaussian energies},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/exact-minimax-homogeneous-complex-gaussian-energies/v/2}
}

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