Functional Estimation for Density, Regression Models and Processes

by Pons, Odile

• ISBN13: 9789814343732
• ISBN10: 9814343730
• eBook ISBN(s): 9789814343749
• Format: Hardcover
• Copyright: 2011-03-31
• Publisher: Textstream

Summary

This book presents a unified approach on nonparametric estimators for models of independent observations, jump processes and continuous processes. New estimators are defined and their limiting behavior is studied. From a practical point of view, the book expounds on the construction of estimators for functionals of processes and densities, and provides asymptotic expansions and optimality properties from smooth estimators.It also presents new regular estimators for functionals of processes, compares histogram and kernel estimators of several new estimators for single-index models, and examines the weak convergence of the estimators.

Table of Contents

Preface	p. v
Introduction	p. 1
Estimation of a density	p. 2
Estimation of a regression curve	p. 10
Estimation of functionals of processes	p. 14
Content of the book	p. 19
Kernel estimator of a density	p. 23
Introduction	p. 23
Risks and optimal bandwidths for the kernel estimator	p. 25
Weak convergence	p. 29
Minimax and histogram estimators	p. 33
Estimation of functionals of a density	p. 34
Density of absolutely continuous distributions	p. 37
Hellinger distance between a density and its estimator	p. 39
Estimation of the density under right-censoring	p. 40
Estimation of the density of left-censored variables	p. 42
Kernel estimator for the density of a process	p. 44
Exercises	p. 46
Kernel estimator of a regression function	p. 49
Introduction and notation	p. 49
Risks and convergence rates for the estimator	p. 50
Optimal bandwidths	p. 56
Weak convergence of the estimator	p. 60
Estimation of a regression curve by local polynomials	p. 62
Estimation in regression models with functional variance	p. 64
Estimation of the mode of a regression function	p. 68
Estimation of a regression function under censoring	p. 69
Proportional odds model	p. 70
Estimation for the regression function of processes	p. 71
Exercises	p. 73
Limits for the varying bandwidths estimators	p. 75
Introduction	p. 75
Estimation of densities	p. 76
Estimation of regression functions	p. 81
Estimation for processes	p. 84
Exercises	p. 85
Nonparametric estimation of quantiles	p. 87
Introduction	p. 87
Asymptotics for the quantile processes	p. 89
Bandwidth selection	p. 95
Estimation of the conditional density of Y given X	p. 98
Estimation of conditional quantiles for processes	p. 100
Inverse of a regression function	p. 102
Quantile function of right-censored variables	p. 104
Conditional quantiles with variable bandwidth	p. 105
Exercises	p. 106
Nonparametric estimation of intensities for stochastic processes	p. 107
Introduction	p. 107
Risks and convergences for estimators of the intensity	p. 110
Kernel estimator of the intensity	p. 111
Histogram estimator of the intensity	p. 116
Risks and convergences for multiplicative intensities	p. 118
Models with nonparametric regression functions	p. 119
Models with parametric regression functions	p. 120
Histograms for intensity and regression functions	p. 124
Estimation of the density of duration excess	p. 126
Estimators for processes on increasing intervals	p. 130
Models with varying intensity or regression coefficients	p. 132
Progressive censoring of a random time sequence	p. 135
Exercises	p. 136
Estimation in semi-parametric regression models	p. 137
Introduction	p. 137
Convergence of the estimators	p. 139
Nonparametric regression with a change of variables	p. 143
Exercises	p. 146
Diffusion processes	p. 147
Introduction	p. 147
Estimation for continuous diffusions by discretization	p. 149
Estimation for continuous diffusion processes	p. 154
Estimation of discretely observed diffusions with jumps	p. 158
Continuous estimation for diffusions with jumps	p. 162
Transformations of a non-stationary Gaussian process	p. 164
Exercises	p. 166
Applications to time series	p. 167
Nonparametric estimation of the mean	p. 168
Periodic models for time series	p. 171
Nonparametric estimation of the covariance function	p. 172
Nonparametric transformations for stationarity	p. 174
Change-points in time series	p. 174
Exercises	p. 181
Appendices	p. 183
Appendix A	p. 183
Appendix B	p. 184
Appendix C	p. 184
Appendix D	p. 187
Notations	p. 189
Bibliography	p. 191
Index	p. 197
Table of Contents provided by Ingram. All Rights Reserved.