The (Precise) Convergence Of Some Important Statistics

A new kind of precise asymptotics is studied in Chapter I. On one hand, we apply the new kind of precise asymptotics to U-statistics, and get the law of logarithm and the law of iterated logarithm. On the other hand, We discuss the precise asymptotics of the strong law of large number and the law of logarithm in linear process assuming innovation variables are φ-mixing, and get the optimal under the mixing coefficient.In Chapter Ⅱ-Section One, we study nonparametric functional regression estimation for α-mixing processes and get strong and weak consistency under more general conditions. In Section Two, we construct confidence intervals for nonparametric functional data analysis using empirical likelihood. It is obviously that the property of confidence intervals are better than that constructed by asymptotic Normal.