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

Bouzebda, Salim; Souddi, Youssouf; Madani, Fethi

doi:10.3390/math12030448

Open AccessArticle

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

by

Salim Bouzebda

^1,*

,

Youssouf Souddi

²

and

Fethi Madani

²

¹

Laboratoire de Mathématiques Appliquées de Compiègne (L.M.A.C.), Université de Technologie de Compiègne, 60200 Compiègne, France

²

Laboratory of Stochastic Models, Statistics and Applications, University of Saida-Dr. Moulay Tahar, P.O. Box 138 EN-NASR, Saïda 20000, Algeria

^*

Author to whom correspondence should be addressed.

Mathematics 2024, 12(3), 448; https://doi.org/10.3390/math12030448

Submission received: 18 January 2024 / Revised: 25 January 2024 / Accepted: 29 January 2024 / Published: 30 January 2024

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Abstract

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.

Keywords: conditional distribution; small ball probability; missing at random; empirical process; ergodic functional data; semi-metric space; covering number

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MDPI and ACS Style

Bouzebda, S.; Souddi, Y.; Madani, F. Weak Convergence of the Conditional Set-Indexed Empirical Process for Missing at Random Functional Ergodic Data. Mathematics 2024, 12, 448. https://doi.org/10.3390/math12030448

AMA Style

Bouzebda S, Souddi Y, Madani F. Weak Convergence of the Conditional Set-Indexed Empirical Process for Missing at Random Functional Ergodic Data. Mathematics. 2024; 12(3):448. https://doi.org/10.3390/math12030448

Chicago/Turabian Style

Bouzebda, Salim, Youssouf Souddi, and Fethi Madani. 2024. "Weak Convergence of the Conditional Set-Indexed Empirical Process for Missing at Random Functional Ergodic Data" Mathematics 12, no. 3: 448. https://doi.org/10.3390/math12030448

APA Style

Bouzebda, S., Souddi, Y., & Madani, F. (2024). Weak Convergence of the Conditional Set-Indexed Empirical Process for Missing at Random Functional Ergodic Data. Mathematics, 12(3), 448. https://doi.org/10.3390/math12030448

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

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