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Functional estimation for time series. I: Quadratic convergence properties. (English) Zbl 0893.62024

Summary: Let \({\mathbf Z}= (X_n,Y_n)_{n\in\mathbb{N}^*}\) be a strongly mixing stationary stochastic process. We consider delta-estimates of the density of the marginal distribution of \(X_1\) and of the regression function \(r(x)= \mathbb{E}[Y_1\mid X_1=x]\) for a class of estimators proposed formerly by G. Walter and J. Blum [Ann. Stat. 7, 328-340 (1979; Zbl 0403.62025)]. A finer evaluation of the variance of these estimates may be undertaken thanks to a new covariance inequality. The bounds reach an optimal order (that is of the i.i.d.’s). Optimal bounds for MISE criterion are deduced from this basic result. We also deduce the convergence in law of some quadratic functionals. We examine assumptions of strong dependence and of absolute regularity. Minimax rates are established.

MSC:

62G07 Density estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62J02 General nonlinear regression
60G10 Stationary stochastic processes
60G99 Stochastic processes
62G20 Asymptotic properties of nonparametric inference

Citations:

Zbl 0403.62025
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