Strong Uniform Consistency of Hazard Function with Functional Explicatory Variable in Single Functional Index Model under Censored Data

In this paper we deal with nonparametric estimate of the conditional hazard function, when the covariate is functional. Kernel type estimators for the conditional hazard function of a scalar response variable Y given a Hilbertian random variable X are introduced, where the observations are linked with a single-index structure. We establish the pointwise almost complete convergence and the uniform almost complete convergence (with the rate) of the kernel estimate of this model in various situations, including censored and non-censored data. The rates of convergence emphasize the crucial role played by the small ball probabilities with respect to the distribution of the explanatory functional variable