kNN estimators for time series prediction: a functional partial linear single index model with missing responses and error-prone covariates

This article investigates a functional partial linear single index model for strong $$\alpha$$ α -mixing functional time series data when the responses are missing not at random and the real covariates are observed with measurement errors. We first extend three insertion methods developed for regression models with missing responses in finite dimensions, namely imputation, semiparametric regression surrogate and inverse marginal probability weighted approaches, to functional scenarios when the responses are scalar. Then the attenuation correction method is employed to eliminate the impact of measurement error on model estimation of unknown parameter in linear component, after we completing the missingness of responses by using above insertion methods. Meanwhile, we combine the kNN approach with insertion and attenuation correction approaches to capture the local structure of functional time series data and provide the estimations of unknown operators in the estimation process. The asymptotic properties of unknown parameters in the model are established under some mild assumptions. Furthermore, we make a comparison of three insertion methods, the oracle method and the ignoring method in simulation study and electricity consumption data analysis. All results indicate that our methodology has good performance.