Functional Estimation for Density, Regression Models and Processes

✍ Scribed by Odile Pons

• Publisher: World Scientific Pub Co Inc
• Year: 2023
• Tongue: English
• Leaves: 259
• Edition: 2
• Category: Library

✦ Synopsis

Nonparametric kernel estimators apply to the statistical analysis of independent or dependent sequences of random variables and for samples of continuous or discrete processes. The optimization of these procedures is based on the choice of a bandwidth that minimizes an estimation error and the weak convergence of the estimators is proved. This book introduces new mathematical results on statistical methods for the density and regression functions presented in the mathematical literature and for functions defining more complex models such as the models for the intensity of point processes, for the drift and variance of auto-regressive diffusions and the single-index regression models.This second edition presents minimax properties with Lp risks, for a real p larger than one, and optimal convergence results for new kernel estimators of function defining processes: models for multidimensional variables, periodic intensities, estimators of the distribution functions of censored and truncated variables, estimation in frailty models, estimators for time dependent diffusions, for spatial diffusions and for diffusions with stochastic volatility.

✦ Table of Contents

Contents
Preface of the Second Edition
Preface of the First Edition
About the Author
1. Introduction
1.1 Estimation of a density
1.2 Estimation of a regression curve
1.3 Estimation of functionals of processes
1.4 Content of the book
2. Kernel Estimator of a Density
2.1 Introduction
2.2 Risks and optimal bandwidths for the kernel estimator
2.3 Weak convergence
2.4 Estimation of the density on Rd
2.5 Minimax risk
2.6 Histogram estimator
2.7 Estimation of functionals of a density
2.8 Density of absolutely continuous distributions
2.9 Hellinger distance between a density and its estimator
2.10 Estimation of the density under right-censoring
2.11 Estimation of the density of left-censored variables
2.12 Kernel estimator for the density of a process
2.13 Exercises
3. Kernel Estimator of a Regression Function
3.1 Introduction and notation
3.2 Risks and convergence rates for the estimator
3.3 Optimal bandwidths for derivatives
3.4 Weak convergence of the estimator
3.5 Estimation of a regression function on Rd
3.6 Estimation of a regression curve by local polynomials
3.7 Estimation in regression models with functional variance
3.8 Estimation of the mode of a regression function
3.9 Estimation of a regression function under censoring
3.10 Proportional odds model
3.11 Estimation for the regression function of processes
3.12 Exercises
4. Limits for the Varying Bandwidths Estimators
4.1 Introduction
4.2 Estimation of densities
4.3 Estimation of regression functions
4.4 Estimation for processes
4.5 Exercises
5. Nonparametric Estimation of Quantiles
5.1 Introduction
5.2 Asymptotics for the quantile processes
5.3 Bandwidth selection
5.4 Estimation of the conditional density of Y given X
5.5 Estimation of conditional quantiles for processes
5.6 Inverse of a regression function
5.7 Quantile function of right-censored variables
5.8 Conditional quantiles with varying bandwidth
5.9 Exercises
6. Nonparametric Estimation of Intensities for Stochastic Processes
6.1 Introduction
6.2 Risks and convergences for estimators of the intensity
6.2.1 Kernel estimator of the intensity
6.2.2 Histogram estimator of the intensity
6.3 Risks and convergences for kernel estimators (6.4)
6.3.1 Models with nonparametric regression functions
6.3.2 Models with parametric regression functions
6.4 Histograms for intensity and regression functions
6.5 Estimation of the density of duration excess
6.6 Estimators for processes on increasing intervals
6.7 Conditional intensity under left-truncation
6.8 Conditional intensity under left-truncation and right-censoring
6.9 Models with varying intensity or regression coefficients
6.10 Estimation in nonparametric frailty models
6.11 Bivariate hazard functions
6.12 Progressive censoring of a random time sequence
6.13 Model with periodic baseline intensity
6.14 Exercises
7. Estimation in Semi-parametric Regression Models
7.1 Introduction
7.2 Convergence of the estimators
7.3 Nonparametric regression with a change of variables
7.4 Exercises
8. Diffusion Processes
8.1 Introduction
8.2 Kernel estimation of time dependent diffusions
8.3 Auto-regressive diffusions
8.4 Estimation for auto-regressive diffusions by discretization
8.5 Estimation for continuous diffusion processes
8.6 Estimation of a diffusion with stochastic volatility
8.7 Estimation of an auto-regressive spatial diffusions
8.8 Estimation of discretely observed diffusions with jumps
8.9 Continuous estimation for diffusions with jumps
8.10 Transformations of a nonstationary Gaussian process
8.11 Exercises
9. Applications to Time Series
9.1 Nonparametric estimation of the mean
9.2 Periodic models for time series
9.3 Nonparametric estimation of the covariance function
9.4 Nonparametric transformations for stationarity
9.5 Change-points in time series
9.6 Exercises
10. Appendix
10.1 Appendix A
10.2 Appendix B
10.3 Appendix C
10.4 Appendix D
Notations
Bibliography
Index