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.

Kolmogorov Entropy for Convergence Rate in Incomplete Functional Time Series: Application to Percentile and Cumulative Estimation in High Dimensional Data

“…We shall also use the same symbol τ to denote the induced set transformation, which takes, for example, sets B ∈ B m into sets τB ∈ B m+1 ; for instance, see [52]. The naming of strong mixing in the above definition is more stringent than what is ordinarily referred to (when using the vocabulary of measure preserving dynamical systems) as strong mixing, namely to that lim n→∞ P(A ∩ τ −n B) = P(A)P(B) for any two measurable sets A, B; see, for instance [52,53] and more recent references [54][55][56][57][58][59][60]. Hence, strong mixing implies ergodicity, whereas the inverse is not always true (see, e.g., Remark 2.6 in page 50 in connection with Proposition 2.8 in page 51 in [40]).…”

Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes

“…Following [34,49,50], we will now introduce some basic notations for defining wavelet bases for Hilbert spaces of functions with a few modifications to accommodate our context. In this study, nonlinear, thresholded, wavelet-based estimators are examined.…”

“…In [49], we have considered the wavelet basis for the nonparametric estimation of density and regression functions for continuous functional stationary processes in Hilbert space. We have characterized the mean integrated square errors.…”

“…We have characterized the mean integrated square errors. This research aims to provide the first complete theoretical rationale for wavelet-based functional density and regression function estimation for continuous stationary processes by extending our earlier work [49] to the continuous setting. The employment of wavelet estimators in functional ergodic data frames and the resulting difficulty of establishing the mean integrated square error over suitable decomposition spaces is, to the best of our knowledge, still an unresolved topic in the literature.…”

Weak Convergence of the Conditional Set-Indexed Empirical Process for Missing at Random Functional Ergodic Data

“…Some motivations to consider ergodic dependence structure in the data rather than a mixing one are discussed in Refs. [67,68].…”