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

Didi, Sultana; Bouzebda, Salim

doi:10.3390/math10224356

Cited by 10 publications

(3 citation statements)

References 97 publications

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“…In a future investigation, considering the limiting law of the conditional U-statistics regression estimators based on the delta sequence will be of interest. A natural extension of the present investigation is to consider the serial-dependent setting such as the mixing (see [61,62,122]) or the ergodic processes (see [56,123]). In a future investigation of the functional delta sequence local linear approach estimators, it will be natural to think about the possibility of obtaining an alternative estimator that benefits from the advantages of both methods, the local linear method and the delta sequence approach.…”

Section: Discussionmentioning

confidence: 99%

Uniform Consistency for Functional Conditional U-Statistics Using Delta-Sequences

2022

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U-statistics are a fundamental class of statistics derived from modeling quantities of interest characterized by responses from multiple subjects. U-statistics make generalizations the empirical mean of a random variable X to the sum of all k-tuples of X observations. This paper examines a setting for nonparametric statistical curve estimation based on an infinite-dimensional covariate, including Stute’s estimator as a special case. In this functional context, the class of “delta sequence estimators” is defined and discussed. The orthogonal series method and the histogram method are both included in this class. We achieve almost complete uniform convergence with the rates of these estimators under certain broad conditions. Moreover, in the same context, we show the uniform almost-complete convergence for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship, which is of its own interest. Among the potential applications are discrimination problems, metric learning and the time series prediction from the continuous set of past values.

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Section: Discussionmentioning

confidence: 99%

Uniform Consistency for Functional Conditional U-Statistics Using Delta-Sequences

2022

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

Section: Introductionmentioning

confidence: 99%

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

Bouzebda,

Souddi,

Madani

2024

Mathematics

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This work examines the asymptotic characteristics of a conditional set-indexed empirical process composed of functional ergodic random variables with missing at random (MAR). This paper’s findings enlarge the previous advancements in functional data analysis through the use of empirical process methodologies. These results are shown under specific structural hypotheses regarding entropy and under appealing situations regarding the model. The regression operator’s asymptotic (1−α)-confidence interval is provided for 0<α<1 as an application. Additionally, we offer a classification example to demonstrate the practical importance of the methodology.

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

Cited by 10 publications

References 97 publications

Uniform Consistency for Functional Conditional U-Statistics Using Delta-Sequences

Uniform Consistency for Functional Conditional U-Statistics Using Delta-Sequences

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

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

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