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.

当协变量被观察到有测量误差时，本文开发了 Logistic 回归中的参数估计。在病例对照框架下的逻辑回归中，病例组和对照组之间协变量密度的对数比是回归参数的线性函数。因此，可以根据估计的协变量密度获得 Logistic 回归的集成最小二乘型估计量。当协变量被精确测量时，可以通过核密度估计来有效地估计协变量密度，并且Geng和Sakhanenko（2016）开发了相应的参数估计器。当观察到协变量存在测量误差时，我们通过采用反卷积核密度估计方法提出了最小二乘型参数估计器。当协变量的测量误差是普通光滑时，一致性和渐近正态性成立。仿真研究表明，所提出的估计器在减少误差方差和不平衡病例对照样本的偏差方面具有鲁棒的估计性能。还包括真实的数据应用程序。