Strong uniform consistency rates of conditional quantiles for time series data in the single functional index model

The main objective of this paper is to estimate non-parametrically the quantiles of a conditional distribution when the sample is considered as an $\alpha$-mixing sequence. First of all, a kernel type estimator for the conditional cumulative distribution function ({\em cond-cdf}) is introduced. Afterwards, we give an estimation of the quantiles by inverting this estimated {\em cond-cdf}, the asymptotic properties are stated 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. For that, the whole observed time series has to be split into a set of functional data, and the functional conditional quantile approach can be employed both in foreseeing and building confidence prediction bands