STRONG UNIFORM CONSISTENCY RATES OF CONDITIONAL DENSITY ESTIMATION IN THE SINGLE FUNCTIONAL INDEX MODEL FOR FUNCTIONAL DATA UNDER RANDOM CENSORSHIP

The main objective of this paper is to investigate the estimation of conditional density function based on the single-index model in the censorship model when the sample is considered as an inde-pendent and identically distributed (i.i.d.) random variables. First of all, a kernel type estimator for the conditional density function (cond-df) is introduced. Afterwards, the asymptotic properties are stated when the observations are linked with a single-index structure. The pointwise almost com-plete convergence and the uniform almost complete convergence (with rate) of the kernel estimate of this model are established. As an application the conditional mode in functional single-index model is presented. Finally, a simulation study is carried out to evaluate the performance of this estimate.