Rank-one variance inflation in polynomial estimation of signal parameters with a sign-preserving fractional-power basis
Abstract
The polynomial maximization method estimates signal parameters in non-Gaussian noise from a finite description of the noise by its moments and cumulants, without a density. Known accuracy results assume that these noise characteristics are known, or use a design in which estimating them jointly costs no accuracy. In practice they are estimated from the residuals of a preliminary fit, and the effect of this substitution on accuracy is unknown. We treat the feasible scheme as a two-step estimator and, for a general smooth signal model in a fixed design, find the increment of its asymptotic covariance in closed form: Σfe−Σor=c2(1−gor) Q−1d̅d̅⊤Q−1, a positive semidefinite matrix of rank at most one. Here c2 is the noise variance, gor≤ 1 the oracle variance reduction factor, and Q and d̅ the second-moment matrix and mean of the signal gradient over the design. Estimating the fractional moments that form the weights costs no first-order accuracy, because that block is orthogonal; the whole increment comes from estimating the sign-preserving centering means. The increment lies along the single direction Q−1d̅, so linear combinations of the parameters orthogonal to it keep oracle accuracy. In the trace metric the variance reduction factor becomes gfe=gor+(1−gor)RD, where RD=|Q−1d̅|2/tr Q−1 depends on the design alone. We give four sufficient conditions for first-order oracle equivalence; noise symmetry is not among them. Monte Carlo experiments with a harmonic signal confirm the direction and size of the increment. A covariance estimate that ignores the increment undercovers where non-Gaussianity gives the largest gain, symmetric noise included.
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BibTeX
@misc{zabolotnii2026rankone,
title = {{Rank-one variance inflation in polynomial estimation of signal parameters with a sign-preserving fractional-power basis}},
author = {Serhii Zabolotnii},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05511},
eprint = {2610.05511},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05511},
}
RIS
TY - UNPB
TI - Rank-one variance inflation in polynomial estimation of signal parameters with a sign-preserving fractional-power basis
AU - Serhii Zabolotnii
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.05511
UR - https://arxiv.org/abs/2610.05511
AB - The polynomial maximization method estimates signal parameters in non-Gaussian noise from a finite description of the noise by its moments and cumulants, without a density. Known accuracy results assume that these noise characteristics are known, or use a design in which estimating them jointly costs no accuracy. In practice they are estimated from the residuals of a preliminary fit, and the effect of this substitution on accuracy is unknown. We treat the feasible scheme as a two-step estimator and, for a general smooth signal model in a fixed design, find the increment of its asymptotic covariance in closed form: Σ_fe−Σ_or=c_2(1−g_or) Q^−1d̅d̅^⊤Q^−1, a positive semidefinite matrix of rank at most one. Here c_2 is the noise variance, g_or≤ 1 the oracle variance reduction factor, and Q and d̅ the second-moment matrix and mean of the signal gradient over the design. Estimating the fractional moments that form the weights costs no first-order accuracy, because that block is orthogonal; the whole increment comes from estimating the sign-preserving centering means. The increment lies along the single direction Q^−1d̅, so linear combinations of the parameters orthogonal to it keep oracle accuracy. In the trace metric the variance reduction factor becomes g_fe=g_or+(1−g_or)R_D, where R_D=|Q^−1d̅|^2/tr Q^−1 depends on the design alone. We give four sufficient conditions for first-order oracle equivalence; noise symmetry is not among them. Monte Carlo experiments with a harmonic signal confirm the direction and size of the increment. A covariance estimate that ignores the increment undercovers where non-Gaussianity gives the largest gain, symmetric noise included.
ER -