Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample $ (X_t, Y_t, \zeta _t)_{t\in [0,T]} $ (Xt,Yt,ζt)t∈[0,T] copies of a continuous-time stationary and ergodic process $ (X, Y, \zeta ) $ (X,Y,ζ). The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process $ \zeta = 1 $ ζ=1 and missing whenever $ \zeta = 0 $ ζ=0. Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.