Investigating On Robust Estimate For Long-dependent Functional Data

Functional data is a function form of the data, its greatest feature is a function of the data. In the current data analysis and processing, if the observation point in time when very close, the data will be presented in the data space, a kind of functional characteristics. For this reason, the functional data is widely used in criminology, economics, archeology, and neurophysiology, etc., has led more scholars to join today to the functional data in the field of study. Nonparametric estimation is an important mathematical branch of statistics, the statistical model requires a broad, can use the general information of data. The method is suitable for a wide range, good stability and good large sample properties. While the method of kernel estimation is the core of the regression of nonparametric estimation, it can very sensitive to outliers, does not have a certain degree of interference immunity. Robust estimation is to solve the traditional kernel estimator proposed by poor anti-interference. Weakly dependent process that can be widely used, but may not contain all of the stationary data, and these occasions of long-range dependent have their wide applications in time series, so this article investigating in robust estimate for long-dependent functional has a practical significance.This work has two parts, first part introduces the long memory of the conditions of the nonparametric regression function M-estimation, has been the result of its convergence in probability, and the speed of convergence in probability; The second part focuses on the distribution function of the non-parametric M-estimation, as has also been the result of convergence in probability, and the corresponding speed...