Bias Correction for Semiparametric Regression Models

Abstract

We consider a broad class of semiparametric regression models in which the conditional distribution of the response takes the form $f\{Y|\bf{x}^{\rm T}\boldsymbol{\beta}+m(z), \phi\}$, which is known up to a parametric component $\boldsymbol{\beta}$ of diverging dimension $p$, a smooth function $m(\cdot)$, and a dispersion parameter $\phi$. Existing semiparametric literature on such models has primarily focused on semiparametric efficiency for $\boldsymbol{\beta}$, typically treating $\phi$ and $m(\cdot)$ as nuisances and largely ignoring their finite-sample bias. However, the finite-sample bias of standard estimators can be substantial (especially when $p$ is large relatively to $n$ and/or dispersion is high) and can seriously undermine inference for $\boldsymbol{\beta}$. Moreover, $\phi$ is often of direct scientific interest and requires accurate estimation. To address this gap, we propose SABRE, a simulation-based bias correction framework for this broad semiparametric model class. We establish asymptotic properties of SABRE for the subclass of generalized partially linear models, where bias reduction for $\boldsymbol{\beta}$ and $\phi$ can be achieved without inflating variance, and we outline how the underlying principle may be adapted more generally. Comprehensive simulation studies and a real-data application on early-stage diabetes demonstrate the empirical effectiveness of SABRE in reducing bias and improving inference.