Strong uniform consistency rates of conditional quantile estimation in the single functional index model under random censorship

The main objective of this paper is to non-parametrically estimate the quantiles of a conditional distribution in the censorship model when the sample is considered as an -mixing sequence. First of all, a kernel type estimator for the conditional cumulative distribution function (cond-cdf) is introduced. Afterwards, we estimate the quantiles by inverting this estimated cond-cdf and state the asymptotic properties when the observations are linked with a single-index structure. The pointwise almost complete convergence and the uniform almost complete convergence (with rate) of the kernel estimate of this model are established. This approach can be applied in time series analysis.