Rank-one variance inflation in polynomial estimation of signal parameters with a sign-preserving fractional-power basis

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: ${\mathrm{\Sigma }}_{\mathrm{f}\mathrm{e}}-{\mathrm{\Sigma }}_{\mathrm{o}\mathrm{r}}=c_2(1-g_{\mathrm{o}\mathrm{r}})Q^{-1}\bar{d}{\bar{d}}^{\mathrm{\top }}{Q}^{-1}$, a positive semidefinite matrix of rank at most one. Here $c_2$ is the noise variance, $g_{\mathrm{o}\mathrm{r}}\le 1$ 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 $g_{\mathrm{f}\mathrm{e}}=g_{\mathrm{o}\mathrm{r}}+(1-g_{\mathrm{o}\mathrm{r}})R_D$, where $R_D=\mathrm{\parallel }{Q}^{-1}\bar{d}{\mathrm{\parallel }}^2\mathrm{/}\mathrm{t}\mathrm{r}{Q}^{-1}$ 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.