Ebook: Functional Estimation for Density, Regression Models and Processes

This volume discusses the extended stochastic integral (ESI) (or Skorokhod-Hitsuda integral) and its relation to the logarithmic derivative of differentiable measure along the vector or operator field. In addition, the theory of surface measures and the theory of heat potentials in infinite-dimensional spaces are discussed. These theories are closely related to ESI. It starts with an account of classic stochastic analysis in the Weiner spaces; and then discusses in detail the ESI for the Weiner measure including properties of this integral understood as a process. Moreover, the ESI with a nonrandom kernel is investigated. Some chapters are devoted to the definition and the investigation of properties of the ESI for Gaussian and differentiable measures. Surface measures in Banach spaces and heat potentials theory in Hilbert space are also discussed 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, compares several new estimators for single-index models, and it examines the weak convergence of the estimators.  Read more... 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. Minimax and histogram estimators. 2.5. Estimation of functionals of a density. 2.6. Density of absolutely continuous distributions. 2.7. Hellinger distance between a density and its estimator. 2.8. Estimation of the density under right-censoring. 2.9. Estimation of the density of left-censored variables. 2.10. Kernel estimator for the density of a process. 2.11. 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. 3.4. Weak convergence of the estimator. 3.5. Estimation of a regression curve by local polynomials. 3.6. Estimation in regression models with functional variance. 3.7. Estimation of the mode of a regression function. 3.8. Estimation of a regression function under censoring. 3.9. Proportional odds model. 3.10. Estimation for the regression function of processes. 3.11. 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 variable bandwidth. 5.9. Exercises -- 6. Nonparametric estimation of intensities for stochastic processes. 6.2. Introduction. 6.2. Risks and convergences for estimators of the intensity. 6.3. Risks and convergences for multiplicative intensities. 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. Models with varying intensity or regression coefficients. 6.8. Progressive censoring of a random time sequence. 6.9. 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. Estimation for continuous diffusions by discretization. 8.3. Estimation for continuous diffusion processes. 8.4. Estimation of discretely observed diffusions with jumps. 8.5. Continuous estimation for diffusions with jumps. 8.6. Transformations of a non-stationary Gaussian process. 8.7. 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