Nonparametric statistics for stochastic processes
type: Item
instance of: scholarly article
title: Nonparametric statistics for stochastic processes
zbMATH Open document ID: 0857.62081
published in: Lecture Notes in Statistics
publication date: 1996-06-09T00:00:00Z
review text: The book contains a large number of known and recent results on nonparametric estimation of densities and regression functions for stochastic processes in discrete and continuous time. Kernel estimates for regression functions are applied to the nonparametric prediction problem. To describe the dependence structure in a stochastic process the author used different mixing concepts and applied coupling techniques. The corresponding inequalities for mixing processes are presented in Chapter 1. Chapter 2 contains results on density estimation for discrete-time processes where the optimal asymptotics are obtained for quadratic errors. Furthermore conditions for uniform almost sure convergence and asymptotic normality of kernel estimators are established. Nonparametric estimators for the regression function in a model with random regressors are studied in Chapter 3. The optimal asymptotics are obtained for mean square error. The results are applied to the prediction of Markov processes of order \(k\) and other processes. Chapters 4 and 5 contain results on estimators of density and regression functions and applications to the prediction problem for continuous-time processes. In contrary to the discrete time case for continuous time processes with irregular paths the effect of superoptimality appears. This book contains almost all important classical and a large number of new results for nonparametric estimation if the sample is a stochastic process. It can be recommended not only as a very clear introduction. One can find results and hints for many problems. Even specialists will find several new results.
Mathematics Subject Classification ID: 62M09
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62M20
zbMATH DE Number: 885949
zbMATH Keywords: nonparametric prediction
zbMATH Keywords: dependence structure
zbMATH Keywords: coupling techniques
zbMATH Keywords: inequalities
zbMATH Keywords: mixing processes
zbMATH Keywords: density estimation
zbMATH Keywords: discrete-time processes
zbMATH Keywords: quadratic errors
zbMATH Keywords: uniform almost sure convergence
zbMATH Keywords: asymptotic normality
zbMATH Keywords: kernel estimators
zbMATH Keywords: random regressors
zbMATH Keywords: optimal asymptotics
zbMATH Keywords: mean square error
zbMATH Keywords: prediction of Markov processes
zbMATH Keywords: regression functions
zbMATH Keywords: continuous-time processes
zbMATH Keywords: irregular paths
zbMATH Keywords: superoptimality
author: Denis Bosq
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 885949