Wavelet Density and Regression Estimators for Functional Stationary and Ergodic Data: Discrete Time

Abstract

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.

Keywords

• multivariate regression estimation
• multivariate density estimation
• stationarity
• ergodicity
• rates of strong convergence
• wavelet-based estimators