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Poisson and Gaussian approximation of weighted local empirical processes

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  • Einmahl, John H. J.

Abstract

We consider the local empirical process indexed by sets, a substantial generalization of the well-studied uniform tail empirical process. We show that the weak limit of weighted versions of this process is Poisson under certain conditions, whereas it is Gaussian in other situations. Our main theorems provide many new results as well as a unified approach to a number of asymptotic distributional results for weighted empirical processes, which up to now appeared to be isolated facts. Our results have applications in 'local' statistical procedures; we will, in particular, show their usefulness in multivariate extreme value theory.

Suggested Citation

  • Einmahl, John H. J., 1997. "Poisson and Gaussian approximation of weighted local empirical processes," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 31-58, October.
    • Handle: RePEc:eee:spapps:v:70:y:1997:i:1:p:31-58
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    1. Einmahl, John H.J. & de Haan, Laurens & Sinha, Ashoke Kumar, 1997. "Estimating the spectral measure of an extreme value distribution," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 143-171, October.
    2. Einmahl, J.H.J., 1987. "Multivariate empirical processes," Other publications TiSEM 4d74fa6b-5281-48ea-aa4d-5, Tilburg University, School of Economics and Management.
    3. Einmahl, J.H.J., 1996. "Extension to higher dimensions of the Jaeschke-Eicker result on the standardized empirical process," Other publications TiSEM e4026f4e-14f7-4c80-9f72-2, Tilburg University, School of Economics and Management.
    4. Csörgo, Miklós & Mason, David M., 1985. "On the asymptotic distribution of weighted uniform empirical and quantile processes in the middle and on the tails," Stochastic Processes and their Applications, Elsevier, vol. 21(1), pages 119-132, December.
    5. Mason, David M., 1983. "The asymptotic distribution of weighted empirical distribution functions," Stochastic Processes and their Applications, Elsevier, vol. 15(1), pages 99-109, June.
    6. Einmahl, J.H.J. & Ruymgaart, F.H., 1995. "Tail processes under heavy random censorship with applications," Other publications TiSEM a77f6162-4e20-4e5f-8250-9, Tilburg University, School of Economics and Management.
    7. Einmahl, J.H.J. & Mason, D.M., 1988. "Laws of the iterated logarithm in the tails for weighted uniform empirical processes," Other publications TiSEM d5a5a8d5-c060-4344-9675-2, Tilburg University, School of Economics and Management.
    8. Einmahl, J. H.J., 1992. "The almost sure behavior of the weighted empirical process and the LIL for the weighted tail empirical process," Other publications TiSEM 5520438c-0aea-424b-b2c4-2, Tilburg University, School of Economics and Management.
    9. Einmahl, J. H.J. & Dekkers, A. L.M. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Other publications TiSEM 81970cb3-5b7a-4cad-9bf6-2, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Einmahl, J.H.J. & Krajina, A. & Segers, J., 2011. "An M-Estimator for Tail Dependence in Arbitrary Dimensions," Other publications TiSEM 27508aa0-9825-4d9e-b1f4-1, Tilburg University, School of Economics and Management.
    2. Einmahl, J.H.J. & Deheuvels, P., 2000. "Functional limit laws for the increments of Kaplan-Meier product-limit processes and applications," Other publications TiSEM ac9bbdc0-62f8-4b48-9a84-1, Tilburg University, School of Economics and Management.
    3. Chan, Ngai-Hang & Lee, Thomas C.M. & Peng, Liang, 2010. "On nonparametric local inference for density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 509-515, February.
    4. Claudia Klüppelberg & Gabriel Kuhn & Liang Peng, 2008. "Semi‐Parametric Models for the Multivariate Tail Dependence Function – the Asymptotically Dependent Case," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(4), pages 701-718, December.
    5. Einmahl, J.H.J. & Segers, J.J.J., 2008. "Maximum Empirical Likelihood Estimation of the Spectral Measure of an Extreme Value Distribution," Discussion Paper 2008-42, Tilburg University, Center for Economic Research.
    6. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximation of suprema of empirical processes," CeMMAP working papers CWP44/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. John H. J. Einmahl & Fan Yang & Chen Zhou, 2021. "Testing the Multivariate Regular Variation Model," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(4), pages 907-919, October.
    8. D’Haultfœuille, Xavier & Hoderlein, Stefan & Sasaki, Yuya, 2023. "Nonparametric difference-in-differences in repeated cross-sections with continuous treatments," Journal of Econometrics, Elsevier, vol. 234(2), pages 664-690.
    9. Estate Khmaladze & Wolfgang Weil, 2018. "Fold-up derivatives of set-valued functions and the change-set problem: A Survey," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(1), pages 1-38, February.
    10. Einmahl, J.H.J. & Khmaladze, E.V., 2007. "Central Limit Theorems For Local Emprical Processes Near Boundaries of Sets," Discussion Paper 2007-66, Tilburg University, Center for Economic Research.
    11. Einmahl, John H.J. & de Haan, Laurens & Sinha, Ashoke Kumar, 1997. "Estimating the spectral measure of an extreme value distribution," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 143-171, October.
    12. Jan Beirlant & John H. J. Einmahl, 2010. "Asymptotics for the Hirsch Index," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(3), pages 355-364, September.
    13. Uwe Einmahl & David M. Mason, 2000. "An Empirical Process Approach to the Uniform Consistency of Kernel-Type Function Estimators," Journal of Theoretical Probability, Springer, vol. 13(1), pages 1-37, January.
    14. Krajina, A., 2010. "An M-estimator of multivariate tail dependence," Other publications TiSEM 66518e07-db9a-4446-81be-c, Tilburg University, School of Economics and Management.
    15. Chiapino, Mael & Sabourin, Anne & Segers, Johan, 2018. "Identifying groups of variables with the potential of being large simultaneously," LIDAM Discussion Papers ISBA 2018006, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    16. Einmahl, John & Krajina, Andrea, 2023. "Empirical Likelihood Based Testing for Multivariate Regular Variation," Other publications TiSEM 261583f5-c571-48c6-8cea-9, Tilburg University, School of Economics and Management.
    17. Einmahl, J.H.J. & de Haan, L.F.M. & Piterbarg, V.I., 2001. "Nonparametric estimation of the spectral measure of an extreme value distribution," Other publications TiSEM c3485b9b-a0bd-456f-9baa-0, Tilburg University, School of Economics and Management.
    18. Estate Khmaladze & Wolfgang Weil, 2008. "Local empirical processes near boundaries of convex bodies," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(4), pages 813-842, December.
    19. Einmahl, John & Krajina, Andrea, 2023. "Empirical Likelihood Based Testing for Multivariate Regular Variation," Discussion Paper 2023-001, Tilburg University, Center for Economic Research.
    20. Dümbgen, Lutz & Wellner, Jon A. & Wolff, Malcolm, 2016. "A law of the iterated logarithm for Grenander’s estimator," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3854-3864.

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