Functional Estimation for Density, Regression Models and Processes
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.
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2 Kernel estimator of a density
3 Kernel estimator of a regression function
4 Limits for the varying bandwidths estimators
5 Nonparametric estimation of quantiles
6 Nonparametric estimation of intensities for stochastic processes
7 Estimation in semiparametric regression models
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approximated bandwidth h bias and variance bounded bounded function Brownian motion centered Gaussian process changeIpoint class C2 conditional density conditional distribution conditional distribution function Conditions 2.1 consistent estimator continuous convergence rate converges in probability converges to zero converges uniformly converges weakly covariances zero deﬁned denoted density estimator density f empirical distribution function ergodicity estimated by smoothing expansion ﬁnite ﬁrst ﬁxed function F FY|X Gaussian variable global bandwidths hazard function Hellinger distance histogram implies intensity interval inverse Kaplan-Meier estimator kernel estimator Lemma limit martingale mean squared error mean zero metric space modiﬁed monotone nonparametric estimator nonparametric regression observations optimal bandwidth parameter point process process with mean Proof Proposition 3.1 random regression model respect sample sample-path satisﬁes Statist tends to inﬁnity tends to zero term Theorem uniformly consistent variance function vector weak convergence zero and variance