Generalized regression operator estimation for continuous time functional data processes with missing at random response
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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, ζ_t)_t∈ ℝ^+ copies of a continuous-time stationary ergodic process (X, Y, ζ). The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when ζ= 1 and missing whenever ζ = 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.
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