Asymptotic Properties Of Nearest Neighbor Estimation For Functional Data

In this thesis, we consider the asymptotic properties of nearest neighbor estimation for functional data. We first review some background knowledge about the functional data and nearest neighbor estimation, and then introduce the application of the non-parametric regression. Both nonparametric estimation and testing have been system-atically examined for real-valued i.i.d. or weakly dependent stationary processes. The purpose of the paper is to investigate the nearest neighbor estimation for functional data. We give the assumptions that are necessary in deriving the asymptotic properties of nearest neighbor estimation.Because we improve the assumptions, we get the non-zero bias about the asymp-totic mean of the error. For the empirical data analysis, we consider three different dis-tributions of the errors,standard normal distribution, contaminated errors and Cauchy distribution. The results show that the nearest neighbor estimation is more robust than the kernel estimation. It is more apparent when the errors are heavy-tailed or the data is aberrant. Therefore, when the number of the sample is large and the data has outliers or is aberrant, the nearest neighbor estimator performs a bit better than the kernel estimation.