Two stage smoothing in additive models with missing covariates

This paper considers sparse additive models with missing covariates. The missing mechanism is assumed to be missing at random. The additive components are estimated via a two stage method. First, the penalized weighted least squares method is used. The weight is the inverse of the selection probability, which is the probability of observing covariates. As the penalty, we utilize the adaptive group lasso to distinguish between the zero and the nonzero components. Thus, the penalty is used to investigate the sparse structure and the weight reflects the missing structure. The estimator obtained from the penalized weighted least squares method is denoted by the first stage estimator (FSE). We show the sparsity and consistency properties of the FSE. However, the asymptotic distribution of the FSE of the nonzero components is not derived as it is difficult. Therefore for each nonzero component, we apply the penalized spline methods for univariate regression with the residual of the FSE of other component. The asymptotic normality of the second stage estimator is shown. To confirm the performance of the proposed estimator, simulation studies and a real data application are implemented.