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: ${\Sigma }_{\mathrm{fe}}-{\Sigma }_{\mathrm{or}}=c_2(1-g_{\mathrm{or}})Q^{-1}\bar{d}{\bar{d}}^{\top }{Q}^{-1}$, a positive semidefinite matrix of rank at most one. Here $c_2$ is the noise variance, $g_{\mathrm{or}}\le 1$ the oracle variance reduction factor, and $Q$ and $\bar{d}$ 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 $Q^{-1}\bar{d}$, so linear combinations of the parameters orthogonal to it keep oracle accuracy. In the trace metric the variance reduction factor becomes $g_{\mathrm{fe}}=g_{\mathrm{or}}+(1-g_{\mathrm{or}})R_D$, where $R_D=\Vert Q^{-1}\bar{d}{\Vert }^2/\mathrm{tr}{Q}^{-1}$ 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.

arXiv comment29 pages, 3 figures, 8 tables. v2: corrected Sections 1.2 and 5.5 (relation to arXiv:2606.11421 is a shared mechanism, not a special case). Code and stored results reproducing every number, table and figure: https://github.com/SZabolotnii/Ku-PMM-Rank-One-code-supplement