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Strong approximations for the p-fold integrated empirical process with applications to statistical tests

Author

Listed:
  • Sergio Alvarez-Andrade

    (Université de Technologie de Compiègne)

  • Salim Bouzebda

    (Université de Technologie de Compiègne)

  • Aimé Lachal

    (Université de Lyon)

Abstract

The main purpose of this paper is to investigate the strong approximation of the p-fold integrated empirical process, p being a fixed positive integer. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (Z Wahrscheinlichkeitstheorie und Verw Gebiete 32:111–131, 1975)’s results. Applications include the two-sample testing procedures together with the change-point problems. Finally, simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the integrated empirical processes.

Suggested Citation

  • Sergio Alvarez-Andrade & Salim Bouzebda & Aimé Lachal, 2018. "Strong approximations for the p-fold integrated empirical process with applications to statistical tests," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(4), pages 826-849, December.
    • Handle: RePEc:spr:testjl:v:27:y:2018:i:4:d:10.1007_s11749-017-0572-0
    DOI: 10.1007/s11749-017-0572-0
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    References listed on IDEAS

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    1. Alexander Aue & Lajos Horváth, 2013. "Structural breaks in time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(1), pages 1-16, January.
    2. Szyszkowicz, Barbara, 1994. "Weak convergence of weighted empirical type processes under contiguous and changepoint alternatives," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 281-313, April.
    3. Lajos Horváth & Gregory Rice, 2014. "Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 219-255, June.
    4. Durio, A. & Nikitin, Ya.Yu., 2016. "Local efficiency of integrated goodness-of-fit tests under skew alternatives," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 136-143.
    5. Lajos Horváth & Gregory Rice, 2014. "Rejoinder on: Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 287-290, June.
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    Cited by:

    1. Salim Bouzebda & Youssouf Souddi & Fethi Madani, 2024. "Weak Convergence of the Conditional Set-Indexed Empirical Process for Missing at Random Functional Ergodic Data," Mathematics, MDPI, vol. 12(3), pages 1-22, January.

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