Asymptotics for function derivatives estimators based on stationary and ergodic discrete time processes

Bouzebda, Salim; Chaouch, Mohamed; Biha, Sultana Didi

doi:10.1007/s10463-021-00814-2

Cited by 6 publications

(3 citation statements)

References 64 publications

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“…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]).…”

Section: Lleccdf: Numerical Approximation Of Ccdf-modelmentioning

confidence: 99%

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

Litimein

Fatimah

Bouzebda

et al. 2023

Entropy

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The convergence rate for free-distribution functional data analyses is challenging. It requires some advanced pure mathematics functional analysis tools. This paper aims to bring several contributions to the existing functional data analysis literature. First, we prove in this work that Kolmogorov entropy is a fundamental tool in characterizing the convergence rate of the local linear estimation. Precisely, we use this tool to derive the uniform convergence rate of the local linear estimation of the conditional cumulative distribution function and the local linear estimation conditional quantile function. Second, a central limit theorem for the proposed estimators is established. These results are proved under general assumptions, allowing for the incomplete functional time series case to be covered. Specifically, we model the correlation using the ergodic assumption and assume that the response variable is collected with missing at random. Finally, we conduct Monte Carlo simulations to assess the finite sample performance of the proposed estimators.

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Section: Lleccdf: Numerical Approximation Of Ccdf-modelmentioning

confidence: 99%

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

Litimein

Fatimah

Bouzebda

et al. 2023

Entropy

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“…The lack of research on the general dependence framework for wavelet analysis prompted us to conduct the present study. Several arguments for contemplating an ergodic dependency structure in that data as opposed to a mixing structure are presented in [35][36][37][38][39][40][41][42], where further information on the notion of the ergodic property of processes and examples of such processes are provided. In [43], one of the arguments used to justify the ergodic setting is that establishing ergodic characteristics rather than the mixing condition can be significantly simpler for some classes of processes.…”

Section: Introduction and Motivationsmentioning

confidence: 99%

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

Didi

Bouzebda

2022

Mathematics

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In this study, we look at the wavelet basis for 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 it is 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.

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“…As a result, strong mixing implies ergodicity, whereas the converse is not always true (see, for example, Remark 2.6 on page 50 concerning Proposition 2.8 on page 51 in [35]). Some reasons for considering ergodic dependence structure in data rather than a mixing structure are discussed in [36][37][38][39][40][41][42][43][44], where details on the definition of ergodic property of processes are given, as well as illustrative examples of such processes. One of the arguments used in [45] to justify the ergodic setting is that for certain classes of processes, proving ergodic properties rather than the mixing condition can be much easier.…”

Section: Introductionmentioning

confidence: 99%

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

2022

Self Cite

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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.

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Asymptotics for function derivatives estimators based on stationary and ergodic discrete time processes

Cited by 6 publications

References 64 publications

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

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

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

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

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