Nonparametric Estimator Of Conditional Quantile Under Functional Stationary Ergodic Data

With the improvement of science and technology, the instrument of measurement datais also more and more precise. The ability of people collecting data is increased, thecollected data is also more and more crowed, which is more accurate as a group of dynamicdata, a curve, or surface, or not a group of static data. So the functional data is generallycollected in numerous fields which reflects the complicated change process of objectivewithin continuous time.The functional data analysis is now widely used in many fieldsincluding psychology, meteorology, biology, economics and so on. The estimation ofconditional quantile and its convergence is a important part in the estimation ofnonparametric. Because of its in the economic and financial etc also has a very wide rangeof applications So its also caused the domestic and foreign many scholar’s interests.In this paper, we investigate the nonparametric conditional quantile estimation for thefunctional stationary ergodic data by the martingale and establish the consistency of theestimator under certain conditions.More precisely, in the ergodic data setting, we considerthe conditional quantile where the explanatory variable X taking values in somesemi-metric abstract space and the response variable Y is real-valued.we also give theasymptotic property of conditional distribution function: asymptotic distribution and theconsistency,which extend the related results in the references.