Nonparametric regression estimation for functional stationary ergodic data with missing at random

• We construct the kernel type estimator of the regression operator for functional stationary ergodic data with the responses MAR.
• Some asymptotic properties such as the convergence rate in probability as well as the asymptotic normality of the estimator are obtained under some mild conditions respectively. As an application, the asymptotic (1−ζ1−ζ) confidence interval of the regression operator is also presented for 0<ζ<10<ζ<1.
• A simulation study is carried out to compare the finite sample performance based on mean square error between the classical functional regression in complete case and the functional regression with MAR.
• Because our work is mainly on MAR, how the percentage of missing data affect the prediction result has been shown by Table 1 in this revised version. That is the main contribution of our work.

In this paper, we investigate the asymptotic properties of the estimator for the regression function operator whenever the functional stationary ergodic data with missing at random (MAR) are considered. Concretely, we construct the kernel type estimator of the regression operator for functional stationary ergodic data with the responses MAR, and some asymptotic properties such as the convergence rate in probability as well as the asymptotic normality of the estimator are obtained under some mild conditions respectively. As an application, the asymptotic (1−ζ)(1−ζ) confidence interval of the regression operator is also presented for 0<ζ<10<ζ<1. Finally, a simulation study is carried out to compare the finite sample performance based on mean square error between the classical functional regression in complete case and the functional regression with MAR.