Optimal Experimental Design for the Polynomial Maximization Method: a Higher-Order Reserve Beyond Second-Order Design Theory
Abstract
Optimal designs for regression with asymmetric errors build on the second-order least squares estimator (SLSE) of Wang and Leblanc (2008), whose gain over ordinary least squares (OLS) comes from the error skewness. With i.i.d. errors the optimally weighted SLSE solves the same estimating equation as the degree-two Polynomial Maximization Method (PMM), a family whose degree-S location estimator has asymptotic variance gS≤1 times that of the sample mean. We derive optimal designs for this family when the errors are independent with constant variance but have standardised cumulants of order three and higher that vary with the regressor (cumulant heteroscedasticity). For a known reduction field gS(x), the gS-weighted information matrix MgS obeys the Kiefer-Wolfowitz equivalence theorem. A constant field leaves the OLS-optimal design unchanged; a non-constant field can change it but need not. For symmetric errors with varying kurtosis, the PMM2 and conditional-SLSE designs cannot depart from the OLS-optimal one, whereas a degree-three design can. A stratify-and-pool PMMS estimator attains NCov(θ̂)→σ2MgS−1 at every degree as all strata grow; its per-stratum reduction is below one only for symmetric errors with the symmetry imposed. For a quadratic model on [0,1] with symmetric Beta errors of unit variance and dose-varying shape, the third-order design for the known field has A-efficiency $1.047$ relative to the OLS-optimal design, and $1.001$ under the D-criterion. A two-stage procedure that estimates the field from a one-third pilot builds designs of A-efficiency $1.014−1.020$ at N=600-$2400$; its realised mean-squared-error ratio against the OLS-optimal design ranges from $0.89$ (a loss) to $1.08$.
Related
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- Concave Roofs and Downward Recovery for DR-Submodular Polynomial Maximization
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BibTeX
@misc{zabolotnii2026optimal,
title = {{Optimal Experimental Design for the Polynomial Maximization Method: a Higher-Order Reserve Beyond Second-Order Design Theory}},
author = {Serhii Zabolotnii},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33921},
eprint = {2610.33921},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33921},
}
RIS
TY - UNPB
TI - Optimal Experimental Design for the Polynomial Maximization Method: a Higher-Order Reserve Beyond Second-Order Design Theory
AU - Serhii Zabolotnii
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.33921
UR - https://arxiv.org/abs/2610.33921
AB - Optimal designs for regression with asymmetric errors build on the second-order least squares estimator (SLSE) of Wang and Leblanc (2008), whose gain over ordinary least squares (OLS) comes from the error skewness. With i.i.d. errors the optimally weighted SLSE solves the same estimating equation as the degree-two Polynomial Maximization Method (PMM), a family whose degree-S location estimator has asymptotic variance g_S≤1 times that of the sample mean. We derive optimal designs for this family when the errors are independent with constant variance but have standardised cumulants of order three and higher that vary with the regressor (cumulant heteroscedasticity). For a known reduction field g_S(x), the g_S-weighted information matrix M_g_S obeys the Kiefer-Wolfowitz equivalence theorem. A constant field leaves the OLS-optimal design unchanged; a non-constant field can change it but need not. For symmetric errors with varying kurtosis, the PMM_2 and conditional-SLSE designs cannot depart from the OLS-optimal one, whereas a degree-three design can. A stratify-and-pool PMM_S estimator attains NCov(θ̂)→σ^2M_g_S^−1 at every degree as all strata grow; its per-stratum reduction is below one only for symmetric errors with the symmetry imposed. For a quadratic model on [0,1] with symmetric Beta errors of unit variance and dose-varying shape, the third-order design for the known field has A-efficiency $1.047$ relative to the OLS-optimal design, and $1.001$ under the D-criterion. A two-stage procedure that estimates the field from a one-third pilot builds designs of A-efficiency $1.014−1.020$ at N=600-$2400$; its realised mean-squared-error ratio against the OLS-optimal design ranges from $0.89$ (a loss) to $1.08$.
ER -