Generalized SIMEX Method: Polynomial Approximation for Extrapolation

Measurement error is a common challenge in statistical analysis, often leading to incorrect parameter estimation. To address measurement error effects, the simulation and extrapolation (SIMEX) method is one of the widely used approaches because of its flexibility in model specification and generic scope of application. Key concerns of the SIMEX method include the number of repetitions in generating synthetic data and the choice of extrapolation function to recover the corrected estimates from the error-prone ones. In most of the existing developments, the quadratic function is frequently adopted as the extrapolation function. However, when measurement error effects are tremendously severe, quadratic functions may be suboptimal. In addition, the development of theoretical results of existing methods requires an unrealistic assumption that the true extrapolation function is known. To address those concerns, we propose GSIMEX, extending the SIMEX method by considering a higher-order polynomial function as the extrapolation function, which enables us to approximate the unknown and nonlinear extrapolation function. In addition, to improve the accuracy of the corrected estimator, we integrate subset selection and model averaging strategies. The theoretical results of GSIMEX, including the measurement of the approximation and asymptotic normality of the estimator, are rigorously established. Numerical studies are conducted for justification of validation, which show that GSIMEX is valid for dealing with severe measurement error effects and is flexible in handling different types of data structures and regression models. We analyze the simulated and spatial transcriptomics data to illustrate the usage of GSIMEX.