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Non-Parametric Estimation of the Renewal Function for Multidimensional Random Fields

Author

Listed:
  • Livasoa Andriamampionona

    (Department of Mathematics and Informatics, University of Antananarivo, Antananarivo 101, Madagascar)

  • Victor Harison

    (Department of Mathematics and Informatics, University of Antananarivo, Antananarivo 101, Madagascar)

  • Michel Harel

    (Laboratoire Vie-Santé, UR 24 134, Faculté de Médecine, 2 Av. Martin Luther King, 87025 Limoges, France
    INSPE de Limoges, Université de Limoges, 209 Bd. de Vanteaux, 87000 Limoges, France
    Institut de Mathématiques de Toulouse, UMR CNRS 5219, 31062 Toulouse, France)

Abstract

This paper addresses the almost sure convergence and the asymptotic normality of an estimator of the multidimensional renewal function based on random fields. The estimator is based on a sequence of non-negative independent and identically distributed ( i . i . d . ) multidimensional random fields and is expressed as infinite sums of k -folds convolutions of the empirical distribution function. It is an extension of the work from the case of the two-dimensional random fields to the case of the d -dimensional random fields where d > 2 . This is established by the definition of a “strict order relation”. Concrete applications are given.

Suggested Citation

  • Livasoa Andriamampionona & Victor Harison & Michel Harel, 2024. "Non-Parametric Estimation of the Renewal Function for Multidimensional Random Fields," Mathematics, MDPI, vol. 12(12), pages 1-22, June.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:12:p:1862-:d:1415108
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    References listed on IDEAS

    as
    1. Tran, Lanh Tat, 1992. "Kernel density estimation for linear processes," Stochastic Processes and their Applications, Elsevier, vol. 41(2), pages 281-296, June.
    2. Michel Harel & Jean-François Lenain & Joseph Ngatchou-Wandji, 2016. "Asymptotic behaviour of binned kernel density estimators for locally non-stationary random fields," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(2), pages 296-321, June.
    3. Carbon, Michel & Garel, Bernard & Tran, Lanh Tat, 1997. "Frequency polygons for weakly dependent processes," Statistics & Probability Letters, Elsevier, vol. 33(1), pages 1-13, April.
    4. Livasoa Andriamampionona & Victor Harison & Michel Harel, 2023. "Asymptotic Behavior of a Nonparametric Estimator of the Renewal Function for Random Fields," Mathematics, MDPI, vol. 11(19), pages 1-13, September.
    5. Hallin, Marc & Lu, Zudi & Tran, Lanh T., 2004. "Kernel density estimation for spatial processes: the L1 theory," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 61-75, January.
    6. Biau, Gérard, 2002. "Optimal asymptotic quadratic errors of density estimators on random fields," Statistics & Probability Letters, Elsevier, vol. 60(3), pages 297-307, December.
    7. Tran, Lanh Tat, 1990. "Kernel density estimation on random fields," Journal of Multivariate Analysis, Elsevier, vol. 34(1), pages 37-53, July.
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