A d-dimensional nonparametric additive regression model with dependent observations is considered. Using the marginal integration technique and wavelets methodology, we develop a new adaptive estimator for a component of the additive regression function. Its asymptotic properties are investigated via the minimax approach under the L 2 risk over Besov balls. We prove that it attains a sharp rate of convergence which turns to be the one obtained in the i.i.d. case for the standard univariate regression estimation problem.
In multivariate regression estimation, the rate of convergence depends on the dimension of the regressor. This fact, known as the curse of the dimensionality, motivated several works. The additive model, introduced by Stone (10), offers an efficient response to this problem. In the setting of continuous time processes, using the marginal integration method, we obtain the quadratic convergence rate and the asymptotic normality of the components of the additive model.
In this paper, we establish uniform-in-bandwidth limit laws of the logarithm for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.) estimators of the multivariate regression function under random censorship. A similar result is deduced for estimators of the conditional distribution function. The uniform-inbandwidth consistency for estimators of the conditional density and the conditional hazard rate functions are also derived from our main result. Moreover, the logarithm laws we establish are shown to yield almost sure simultaneous asymptotic confidence bands for the functions we consider. Examples of confidence bands obtained from simulated data are displayed.Key words : censored regression, kernel estimates, laws of the logarithm, inverse probability of censoring weighted estimates.
We consider the nonparametric estimation of the generalised regression
function for continuous time processes with irregular paths when the regressor takes values
in a semimetric space. We establish the mean-square convergence of our estimator with
the same superoptimal rate as when the regressor is real valued.
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