In the present paper, we are mainly concerned with the non parametric estimation of the density as well as the regression function by using orthonormal wavelet bases. We provide the strong uniform consistency properties with rates of these estimators, over compact subsets of , under a general ergodic condition on the underlying processes. We characterize the asymptotic normality of considered wavelet-based estimators, under easily verifiable conditions. The asymptotic properties of these estimators are obtained, by means of the martingale approach.

The main purpose of the present work is to introduce and investigate a simple kernel procedure based on marginal integration that estimates the regression function for stationary and ergodic continuous time processes in the setting of the additive model introduced by Stone ( 1985 Stone, C.J. (1985). Additive regression and other nonparametric models. Ann. Stat. 13(2):689–705. [Crossref], [Web of Science ®] [Google Scholar] ). We obtain the uniform almost sure consistency with exact rate and the asymptotic normality of the kernel-type estimators of the components of the additive model. Asymptotic properties of these estimators are obtained, under mild conditions, by means of martingale approaches. Finally, a general notion of the bootstrapped additive components, constructed by exchangeably weighting sample, is presented.

In this study, we look at the wavelet basis for the nonparametric estimation of density and regression functions for continuous functional stationary processes in Hilbert space. The mean integrated squared error for a small subset is established. We employ a martingale approach to obtain the asymptotic properties of these wavelet estimators. These findings are established under rather broad assumptions. All we assume about the data is that they are ergodic, but beyond that, we make no assumptions. In this paper, the mean integrated squared error findings in the independence or mixing setting were generalized to the ergodic setting. The theoretical results presented in this study are (or will be) valuable resources for various cutting-edge functional data analysis applications. Applications include conditional distribution, conditional quantile, entropy, and curve discrimination.

The derivatives of the probability density or regression functions contain important information concerning a multivariate data set, such as modal regions. Despite this importance, nonparametric estimation of higher-order derivatives of the density or regression functions have received only relatively scant attention. The main purpose of the present work is to investigate general kernel type estimators of function derivatives. We obtain the strong uniform convergence with rate as well as the asymptotic normality for the proposed estimates. We consider the AMISE of kernel derivative estimator which plays a fundamental role for the characterization of the optimal bandwidth. Our results are obtained in the general setting of stationary ergodic processes. Finally, statistical applications include the regression derivatives, the multivariate mode, and the Shannon’s entropy, that are of independent interest.

The nonparametric estimation of density and regression function based on functional stationary processes using wavelet bases for Hilbert spaces of functions is investigated in this paper. The mean integrated square error over adapted decomposition spaces is given. To obtain the asymptotic properties of wavelet density and regression estimators, the Martingale method is used. These results are obtained under some mild conditions on the model; aside from ergodicity, no other assumptions are imposed on the data. This paper extends the scope of some previous results for wavelet density and regression estimators by relaxing the independence or the mixing condition to the ergodicity. Potential applications include the conditional distribution, curve discrimination, and time series prediction from a continuous set of past values.