Calibrated Estimation and Inference 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 $f\{Y|{\mathbf{x}}^T{\hat{\mathbf{I}}}^2+m(z),\ddot{I}•\}$, known up to a parametric component ${\hat{\mathbf{I}}}^2$ of diverging dimension $p$, a smooth function $m(\cdot )$, and a dispersion parameter $\ddot{I}•$. The existing literature on such models has focused on semiparametric efficiency for ${\hat{\mathbf{I}}}^2$, treating $\ddot{I}•$ and $m(\cdot )$ as nuisances and largely ignoring finite-sample bias. Yet this bias can be substantial, particularly when $p$ is large relative to $n$ or the dispersion is high, and it can seriously undermine inference for ${\hat{\mathbf{I}}}^2$; moreover, $\ddot{I}•$ is often of direct scientific interest. We therefore propose SABRE, a general calibration framework for semiparametric estimation and inference, which calibrates an initial estimator against its model-implied expectation under a tractable parametric approximation to the semiparametric model. For generalized partially linear models, we show that SABRE reduces the bias of both ${\hat{\mathbf{I}}}^2$ and $\ddot{I}•$, accommodates a diverging parameter dimension without sparsity, and preserves the first-order variance and semiparametric efficiency of the initial estimator; the joint construction also improves estimation and inference for $m(\cdot )$. Simulation studies and an application to Alzheimer's disease genetics association analysis demonstrate the empirical effectiveness of SABRE in reducing bias and improving inference.