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Mathematics - Probability Theory and Stochastic Processes | Nonparametric Statistics for Stochastic Processes - Estimation and Prediction

Nonparametric Statistics for Stochastic Processes

Estimation and Prediction

Series: Lecture Notes in Statistics, Vol. 110

Bosq, Denis

1996, 1 illus. in color.

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  • About this book

Recently new developments have taken place in the theory of nonpara­ metric statistics for stochastic processes. Optimal asymptotic results have been obtained and special behaviour of estimators and predictors in con­ tinuous time has been pointed out. This book is devoted to these questions. It also gives some indica­ tions about implementation of nonparametric methods and comparaison with parametric ones, including numerical results. Many of the results pre­ sented here are new and have not yet been published, expecially those in Chapters N and V. I am grateful to W. HardIe, Y. Kutoyants, F. Merlevede and G. Oppen­ heim who made important remarks that helped much to improve the text. I am greatly indebted to B. Heliot for her careful reading of the manus­ cript which allowed to ameliorate my english. I also express my gratitude to D. Blanke, L. Cotto and P. Piacentini who read portions of the manuscript and made some useful suggestions. I also thank M. Gilchrist and J. Kimmel for their encouragements. My acknowledgment also goes to M. Carbon, M. Delecroix, B. Milcamps and J .M. Poggi who authorized me to reproduce their numerical results.

Content Level » Research

Keywords » Kernel - Markov process - Variance - mixing - statistics - stochastic processes

Related subjects » Probability Theory and Stochastic Processes

Table of contents 

Synopsis.- 1. The object of the study.- 2. The kernel density estimator.- 3. The kernel regression estimator and the induced predictor.- 4. Mixing processes.- 5. Density estimation.- 6. Regression estimation and Prediction.- 7. Implementation of nonparametric method.- 1. Inequalities for mixing processes.- 1. Mixing.- 2. Coupling.- 3. Inequalities for covariances and joint densities.- 4. Exponential type inequalities.- 5. Some limit theorems for strongly mixing processes.- Notes.- 2. Density estimation for discrete time processes.- 1. Density estimation.- 2. Optimal asymptotic quadratic error.- 3. Uniform almost sure convergence.- 4. Asymptotic normality.- 5. Non regular cases.- Notes.- 3. Regression estimation and prediction for discrete time processes.- 1. Regression estimation.- 2. Asymptotic behaviour of the regression estimator.- 3. Prediction for a stationary Markov process of order k.- 4. Prediction for general processes.- 5. Implementation of nonparametric method.- Notes.- 4. Density estimation for continuous time processes.- 1. The kernel density estimator in continuous time.- 2. Optimal and superoptimal asymptotic quadratic error.- 3. Optimal and superoptimal uniform convergence rates.- 4. Sampling.- Notes.- 5. Regression estimation and prediction in continuous time.- 1. The kernel regression estimator in continuous time.- 2. Optimal asymptotic quadratic error.- 3. Superoptimal asymptotic quadratic error.- 4. Limit in distribution.- 5. Uniform convergence rates.- 6. Sampling.- 7. Nonparametric prediction in continuous time.- Notes.- Appendix—Numerical results.

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