A Study On Mode Estimate For Functional Data

Over the past decades, the functional data, arising from Growth curve analysis,Taxonomy, Biomechanics, Medicine and etc, has been developing onto a highly new level.As related in both subject areas there are a lot of functional data, functional data instatistical inference by more and more attention.This dissertation aims at solving two problems as follows. Firstly, our main aim is to designfunctional a kernel pseudo-estimatorsf%(x)and mode estimatorsθ nof nonparametric functionalvariables X. It is shown that the rate of consistency of the mode estimator for the kernelpseudo-estimator can be determined with the typical smoothness conditions. Indeed, our approach toprove rate of uniform convergence of the kernel pseudo-estimators of some nonparametric models ini.i.d.case and we establish the almost complete convergence rate of the mode estimators which extendthe existing literature related results in i.i.d.case. Secondly, we investigate the rate of uniform completeconvergence for nonparametric estimation of conditional mode with functional variables by theKolmogorov’s entropy and some assumptions in some semi-metric functional space. Under certainconditions, it establish the uniform almost complete convergence rate of conditional density kernelestimator for dependent functional data which extend i.i.d. case to dependence as well as pointwiseconvergence rate to uniform convergence setting.