Calibrated Equilibrium Estimation and Double Selection for High-dimensional Partially Linear Measurement Error Models

In practice, measurement error data is frequently encountered and needs to be handled appropriately. As a result of additional bias induced by measurement error, many existing estimation methods fail to achieve satisfactory performances. This article studies high-dimensional partially linear measurement error models. It proposes a calibrated equilibrium (CARE) estimation method to calibrate the bias caused by measurement error and overcomes the technical difficulty of the objective function unbounded from below in high-dimensional cases due to non-convexity. To facilitate the applications of the CARE estimation method, a bootstrap approach for approximating the covariance matrix of measurement errors is introduced. For the high-dimensional or ultra-high dimensional partially linear measurement error models, a novel multiple testing method, the calibrated equilibrium multiple double selection (CARE–MUSE) algorithm, is proposed to control the false discovery rate (FDR). Under certain regularity conditions, we derive the oracle inequalities for estimation error and prediction risk, along with an upper bound on the number of falsely discovered signs for the CARE estimator. We further establish the convergence rate of the nonparametric function estimator. In addition, FDR and power guarantee for the CARE–MUSE algorithm are investigated under a weaker minimum signal condition, which is insufficient for the CARE estimator to achieve sign consistency. Extensive simulation studies and a real data application demonstrate the satisfactory finite sample performance of the proposed methods.