Nonparametric recursive estimate for right-censored conditional mode function with ergodic functional data

As an extension of some recent works that are essential references for this current contribution, we provide a recursive nonparametric approach to estimate the conditional mode function Theta(x) of the conditional density of a scalar positive random variable Z given a Hilbertian explanatory process X = x; denoted by xi(x) (z), based on the randomly right-censorship model which is the new and main factor here. Our nonparametric model takes into account the fact that the response variable Z referred as a survival time is right-censored by another variable W independent of (Z, X). Afterwards, we establish, under stationary and ergodic conditions, by using an adaptive exponential inequality to this context, some theoretical properties of the resulting estimator including the uniform almost sure convergence (with rates), after establishing the pointwise ones. Finally, an application based on simulated data is conducted to illustrate our results.