IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v195y2023ics0047259x22001348.html
   My bibliography  Save this article

Renewal type bootstrap for increasing degree U-process of a Markov chain

Author

Listed:
  • Soukarieh, Inass
  • Bouzebda, Salim

Abstract

In this paper, we investigate the uniform limit theory for a U-statistic of increasing degree, also called an infinite-degree U-statistic. Infinite-degree U-statistics (IDUS) (or infinite-order U-statistics (IOUS)) are useful tool for constructing simultaneous prediction intervals that quantify the uncertainty of several methods such as subbagging and random forests. The stochastic process based on collections of U-statistics is referred to as a U-process, and if the U-statistic is of infinite-degree, we have an infinite-degree U-process. Heilig and Nolan (2001) provided conditions for the pointwise asymptotic theory for the infinite-degree U-processes. The main purpose here is to extend their findings to the Markovian setting. The second aim is to provide the uniform limit theory for the renewal bootstrap for the infinite-degree U-process, which is of its own interest. The main ingredients are the decoupling technique combined with symmetrization techniques of Heilig and Nolan (2001) to obtain uniform weak law of large numbers and functional central limit theorem for the infinite-degree U-process. The results obtained in this paper are, to our knowledge, the first known results on the infinite-degree U-process in the Markovian setting.

Suggested Citation

  • Soukarieh, Inass & Bouzebda, Salim, 2023. "Renewal type bootstrap for increasing degree U-process of a Markov chain," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    • Handle: RePEc:eee:jmvana:v:195:y:2023:i:c:s0047259x22001348
    DOI: 10.1016/j.jmva.2022.105143
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X22001348
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2022.105143?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Bertail, Patrice & Clemencon, Stephan, 2008. "Approximate regenerative-block bootstrap for Markov chains," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2739-2756, January.
    2. M. Rajarshi, 1990. "Bootstrap in Markov-sequences based on estimates of transition density," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(2), pages 253-268, June.
    3. Fort, G. & Moulines, E. & Priouret, P. & Vandekerkhove, P., 2012. "A simple variance inequality for U-statistics of a Markov chain with applications," Statistics & Probability Letters, Elsevier, vol. 82(6), pages 1193-1201.
    4. Aluísio Pinheiro & Pranab Sen & Hildete Pinheiro, 2011. "A class of asymptotically normal degenerate quasi U-statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(6), pages 1165-1182, December.
    5. Dragan Radulović, 2004. "Renewal type bootstrap for Markov chains," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(1), pages 147-192, June.
    6. Hediger, Simon & Michel, Loris & Näf, Jeffrey, 2022. "On the use of random forest for two-sample testing," Computational Statistics & Data Analysis, Elsevier, vol. 170(C).
    7. Inass Soukarieh & Salim Bouzebda, 2022. "Exchangeably Weighted Bootstraps of General Markov U -Process," Mathematics, MDPI, vol. 10(20), pages 1-42, October.
    8. Shieh, Grace S., 1994. "Infinite order V-statistics," Statistics & Probability Letters, Elsevier, vol. 20(1), pages 75-80, May.
    9. Salim Bouzebda & Issam Elhattab & Boutheina Nemouchi, 2021. "On the uniform-in-bandwidth consistency of the general conditional U-statistics based on the copula representation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 33(2), pages 321-358, April.
    10. Pinheiro, Aluísio & Sen, Pranab Kumar & Pinheiro, Hildete Prisco, 2009. "Decomposability of high-dimensional diversity measures: Quasi-U-statistics, martingales and nonstandard asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1645-1656, September.
    11. Gut, Allan, 1974. "On the moments of some first passage times for sums of dependent random variables," Stochastic Processes and their Applications, Elsevier, vol. 2(1), pages 115-126, January.
    12. Salim Bouzebda & Boutheina Nemouchi, 2020. "Uniform consistency and uniform in bandwidth consistency for nonparametric regression estimates and conditional U-statistics involving functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 32(2), pages 452-509, April.
    13. Salim Bouzebda & Issam Elhattab & Anouar Abdeldjaoued Ferfache, 2022. "General M-Estimator Processes and their m out of n Bootstrap with Functional Nuisance Parameters," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2961-3005, December.
    14. Salim Bouzebda & Thouria El-hadjali, 2020. "Uniform convergence rate of the kernel regression estimator adaptive to intrinsic dimension in presence of censored data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 32(4), pages 864-914, October.
    15. Harel, Michel & Puri, Madan L., 1989. "Weak convergence of the U-statistic and weak invariance of the one-sample rank order statistic for Markov processes and ARMA models," Journal of Multivariate Analysis, Elsevier, vol. 31(2), pages 258-265, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Salim Bouzebda & Amel Nezzal & Tarek Zari, 2022. "Uniform Consistency for Functional Conditional U -Statistics Using Delta-Sequences," Mathematics, MDPI, vol. 11(1), pages 1-39, December.
    2. Salim Bouzebda, 2023. "On Weak Convergence of the Bootstrap Copula Empirical Process with Random Resample Size," Stats, MDPI, vol. 6(1), pages 1-16, February.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Salim Bouzebda & Amel Nezzal & Tarek Zari, 2022. "Uniform Consistency for Functional Conditional U -Statistics Using Delta-Sequences," Mathematics, MDPI, vol. 11(1), pages 1-39, December.
    2. Salim Bouzebda & Yousri Slaoui, 2023. "Nonparametric Recursive Estimation for Multivariate Derivative Functions by Stochastic Approximation Method," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 658-690, February.
    3. Salim Bouzebda & Thouria El-hadjali & Anouar Abdeldjaoued Ferfache, 2023. "Uniform in Bandwidth Consistency of Conditional U-statistics Adaptive to Intrinsic Dimension in Presence of Censored Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1548-1606, August.
    4. Sultana Didi & Salim Bouzebda, 2022. "Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes," Mathematics, MDPI, vol. 10(22), pages 1-37, November.
    5. Bouzebda, Salim & Slaoui, Yousri, 2022. "Nonparametric recursive method for moment generating function kernel-type estimators," Statistics & Probability Letters, Elsevier, vol. 184(C).
    6. Inass Soukarieh & Salim Bouzebda, 2022. "Exchangeably Weighted Bootstraps of General Markov U -Process," Mathematics, MDPI, vol. 10(20), pages 1-42, October.
    7. Salim Bouzebda & Inass Soukarieh, 2022. "Non-Parametric Conditional U -Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design," Mathematics, MDPI, vol. 11(1), pages 1-69, December.
    8. Hildete P. Pinheiro & Rafael P. Maia & Eufrásio A. Lima Neto & Mariana Rodrigues-Motta, 2019. "Zero-one augmented beta and zero-inflated discrete models with heterogeneous dispersion for the analysis of student academic performance," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(4), pages 749-767, December.
    9. Salim Bouzebda & Boutheina Nemouchi, 2023. "Weak-convergence of empirical conditional processes and conditional U-processes involving functional mixing data," Statistical Inference for Stochastic Processes, Springer, vol. 26(1), pages 33-88, April.
    10. Litimein, Ouahiba & Laksaci, Ali & Mechab, Boubaker & Bouzebda, Salim, 2023. "Local linear estimate of the functional expectile regression," Statistics & Probability Letters, Elsevier, vol. 192(C).
    11. repec:ebl:ecbull:v:3:y:2002:i:19:p:1-8 is not listed on IDEAS
    12. Pan, Li & Politis, Dimitris N., 2016. "Bootstrap prediction intervals for Markov processes," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 467-494.
    13. Pan, Li & Politis, Dimitris, 2014. "Bootstrap prediction intervals for Markov processes," University of California at San Diego, Economics Working Paper Series qt7555757g, Department of Economics, UC San Diego.
    14. Manzan, S. & Zerom, D., 2005. "A Multi-Step Forecast Density," CeNDEF Working Papers 05-05, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.
    15. Sherman, Michael & Carlstein, Edward, 2004. "Confidence intervals based on estimators with unknown rates of convergence," Computational Statistics & Data Analysis, Elsevier, vol. 46(1), pages 123-139, May.
    16. Bertail, Patrice & Clemencon, Stephan, 2008. "Approximate regenerative-block bootstrap for Markov chains," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2739-2756, January.
    17. Sudaraka Tholkage & Qi Zheng & Karunarathna B. Kulasekera, 2022. "Conditional Kaplan–Meier Estimator with Functional Covariates for Time-to-Event Data," Stats, MDPI, vol. 5(4), pages 1-17, November.
    18. Hildete P. Pinheiro & Pranab K. Sen & Aluísio Pinheiro & Samara F. Kiihl, 2020. "A nonparametric approach to assess undergraduate performance," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 74(4), pages 538-558, November.
    19. Manzan, Sebastiano & Zerom, Dawit, 2008. "A bootstrap-based non-parametric forecast density," International Journal of Forecasting, Elsevier, vol. 24(3), pages 535-550.
    20. Salim Bouzebda, 2023. "On Weak Convergence of the Bootstrap Copula Empirical Process with Random Resample Size," Stats, MDPI, vol. 6(1), pages 1-16, February.
    21. Zacharia Issa & Blanka Horvath, 2023. "Non-parametric online market regime detection and regime clustering for multidimensional and path-dependent data structures," Papers 2306.15835, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:195:y:2023:i:c:s0047259x22001348. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.