Deconvolution-Based Score Tests in Measurement Error Models

Abstract

Consider a generalized linear model with response $Y$ and scalar predictor $X$. Instead of observing $X$, a surrogate $W = X + Z$ is observed, where $Z$ represents measurement error and is independent of $X$ and $Y$. The efficient score test for the absence of association depends on $m(w) = E(X\mid W = w)$ which is generally unknown. Assuming that the distribution of $Z$ is known, asymptotically efficient tests are constructed using nonparametric estimators of $m(w)$. Rates of convergence for the estimator of $m(w)$ are established in the course of proving efficiency of the proposed test.