Generalized regression operator estimation for continuous time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalized regression function, including conditional cumulative distribution and conditional quantile functions, based on an incomplete sample $(X_t, Y_t, \zeta_t)_{t\in \mathbb{ R}^+}$ copies of a continuous-time stationary ergodic process $(X, Y, \zeta)$. The predictor $X$ is valued in some infinite-dimensional space, whereas the real-valued process $Y$ is observed when $\zeta= 1$ and missing whenever $\zeta = 0$. Pointwise and uniform consistency (with rates) of these estimators as well as a central limit theorem are established. Conditional bias and asymptotic quadratic error are also provided. Asymptotic and bootstrap-based confidence intervals for the generalized regression function are also discussed. A first simulation study is performed to compare the discrete-time to the continuous-time estimations. A second simulation is also conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Finally, it is worth noting that our results are stated under ergodic assumption without assuming any classical mixing conditions.