Parameter estimation for Logistic errors-in-variables regression under case-control studies

The article develops parameter estimation in the Logistic regression when the covariate is observed with measurement error. In Logistic regression under the case-control framework, the logarithmic ratio of the covariate densities between the case and control groups is a linear function of the regression parameters. Hence, an integrated least-square-type estimator of the Logistic regression can be obtained based on the estimated covariate densities. When the covariate is precisely measured, the covariate densities can be effectively estimated by the kernel density estimation and the corresponding parameter estimator was developed by Geng and Sakhanenko (2016). When the covariate is observed with measurement error, we propose the least-square-type parameter estimators by adapting the deconvolution kernel density estimation approach. The consistency and asymptotic normality are established when the measurement error in covariate is ordinary smooth. Simulation study shows robust estimation performance of the proposed estimator in terms of bias reduction against the error variance and unbalanced case-control samples. A real data application is also included.