  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.
