Research On The Asymptotic Properties Of Conditional Quantile Estimator Under Functional Stationary Ergodic Data

In recent years, with the rapid development of computer and information technology,some data which reflects the complicated change process of objective within continuoustime is generally collected in many application fields such as environmental metrology,biostatistics, econometrics, medicine and so on. This kind of data is called the functionaldata. It is one of the hot areas in modern statistics to research the relationship betweenthe variables which have the function features and real valued responsevariables,including constructing the nonparametic estimator of conditional mean,conditional distribution function, conditional density function and conditional quantileand studying its theoretical properties and its application in the related fields.Conditional quantile isn’t subject to outliers compare with conditional mean,so thestatistical inference on it is one of the important questions in nonparametric statistics,it’s the reason why conditional quantile have been concerned by many scholars.On the other hand, because of the complexity of functional data structure, manysamples which we observe in many occasions are not independent of each other buthave a certain dependence, the samples are mostly thought the α-mixing condition inthe existing literature. And although the α-mixing condition is the weskest in thecommon mixing condition,but it can’t cover all the dependent structure,such as theergodic data.Moreover the ergodic framework is more convinent to be verified in timeserises, avoiding the widely used strong mixing. Ergodic hypothesis is oftenencountered in the study of statistical physics,thermodynamics and signal processingand so on.Basing on this background,this paper research the pointwise consistency of theconditional quantile under functional stationary ergodic data.Firstly,this paper isaccording the method of N-W kernel regression estimate to establish the estimators ofthe cumulative distribution function, then using some properties of martingaledifference sequence to get the pointwise convergence rate which is similar with theresult under independent identically distribution.of the cumulative distribution function,finally,we obtained the pointwise convergence rate of the conditional quantile.Ourresults generalize the existing results in the literature.This paper is divided into fourchapters:In the first and second Chapters:Mainly discussed the research background and situation about functional data,and then some preliminaries are giver.In the third Chapter:The pointwise convergence rate of the estimator of thecumulative distribution function and its l order derived function is obtained underfunctional stationary ergodic data, thus gets the pointwise convergence rate of theconditional quantile.In the fourth Chapter:We give a comprehensive summary of the paper.