Polynomial Maximization Method with Fractional Polynomial Basis: A Frequentist Bridge to Bayesian Fractional Polynomials

TL;DR

This paper introduces PMM-FP, a frequentist method extending polynomial maximization to fractional polynomials, offering a computationally efficient bridge to Bayesian fractional polynomial models with proven variance reduction.

Contribution

It develops a new frequentist extension of polynomial maximization for fractional polynomials, including a closed-form variance-reduction coefficient and validation via Monte Carlo.

Findings

Variance-reduction coefficient g_2 approx 0.56 on GBSG residuals

PMM-FP provides a computationally cheap alternative to Bayesian FP modeling

Validated by Monte Carlo simulations with formal proof in Lean 4

Abstract

Fractional polynomials are widely used for dose-response modelling, and recent Bayesian fractional polynomial work has renewed interest in this finite model class. We propose PMM-FP, a frequentist extension of Kunchenko's polynomial maximization method to fractional-polynomial bases, developed in two parallel tracks for positive and full FP power sets under appropriate moment conditions. The main result is the closed-form variance-reduction coefficient g_2=1-gamma_3^2/(2+gamma_4) relative to OLS-FP for asymmetric non-Gaussian errors, formalised in Lean 4 and validated by Monte Carlo. On GBSG residuals, gamma_3=-1.74, gamma_4=4.91, g_2 approx 0.56: an expected standard-error gain. PMM-FP is a computationally cheap frequentist bridge to Bayesian FP modelling.