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Minimum volume sets and generalized quantile processes

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  • Polonik, Wolfgang

Abstract

Bahadur-Kiefer approximations for generalized quantile processes as defined in Einmahl and Mason (1992) are given which generalize results for the classical one-dimensional quantile processes. An as application we consider the special case of the volume process of minimum volume sets in classes of subsets of the d-dimensional Euclidean space. Minimum volume sets can be used as estimators of level sets of a density and might be useful in cluster analysis. The volume of minimum volume sets itself can be used for robust estimation of scale. Consistency results and rates of convergence for minimum volume sets are given. Rates of convergence of minimum volume sets can be used to obtain Bahadur-Kiefer approximations for the corresponding volume process and vice versa. A generalization of the minimum volume approach to non-i.i.d. problems like regression and spectral analysis of time series is discussed.

Suggested Citation

  • Polonik, Wolfgang, 1997. "Minimum volume sets and generalized quantile processes," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 1-24, July.
    • Handle: RePEc:eee:spapps:v:69:y:1997:i:1:p:1-24
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    1. Dahlhaus, Rainer, 1988. "Empirical spectral processes and their applications to time series analysis," Stochastic Processes and their Applications, Elsevier, vol. 30(1), pages 69-83, November.
    2. Einmahl, J. H.J. & Mason, D.M., 1992. "Generalized quantile processes," Other publications TiSEM b2a76bac-045d-457f-869f-d, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Berthet, Philippe & Einmahl, John, 2020. "Cube Root Weak Convergence of Empirical Estimators of a Density Level Set," Other publications TiSEM 69103be2-c944-4ca1-b9e1-2, Tilburg University, School of Economics and Management.
    2. Moritz Herrmann & Fabian Scheipl, 2021. "A Geometric Perspective on Functional Outlier Detection," Stats, MDPI, vol. 4(4), pages 1-41, November.
    3. Jianqing Fan & Mingjin Wang & Qiwei Yao, 2008. "Modelling multivariate volatilities via conditionally uncorrelated components," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(4), pages 679-702, September.
    4. Ren, Qunshu & Mojirsheibani, Majid, 2008. "Nonparametric estimation of level sets under minimal assumptions," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 3029-3033, December.
    5. Burman, Prabir & Polonik, Wolfgang, 2009. "Multivariate mode hunting: Data analytic tools with measures of significance," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1198-1218, July.
    6. Polonik, Wolfgang & Wang, Zailong, 2010. "PRIM analysis," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 525-540, March.
    7. Pavlides, Marios G. & Wellner, Jon A., 2012. "Nonparametric estimation of multivariate scale mixtures of uniform densities," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 71-89.
    8. Di, J. & Kolaczyk, E., 2010. "Complexity-penalized estimation of minimum volume sets for dependent data," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 1910-1926, October.
    9. Baíllo, Amparo, 2003. "Total error in a plug-in estimator of level sets," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 411-417, December.
    10. Hlubinka, Daniel & Šiman, Miroslav, 2013. "On elliptical quantiles in the quantile regression setup," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 163-171.
    11. Daniel Hlubinka & Miroslav Šiman, 2015. "On generalized elliptical quantiles in the nonlinear quantile regression setup," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 249-264, June.
    12. Baíllo, Amparo, 2003. "Total error in a plug-in estimator of level sets," DES - Working Papers. Statistics and Econometrics. WS ws032806, Universidad Carlos III de Madrid. Departamento de Estadística.
    13. J Morio & R Pastel, 2012. "Plug-in estimation of d-dimensional density minimum volume set of a rare event in a complex system," Journal of Risk and Reliability, , vol. 226(3), pages 337-345, June.
    14. Paula Saavedra-Nieves & Rosa M. Crujeiras, 2022. "Nonparametric estimation of directional highest density regions," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 16(3), pages 761-796, September.
    15. Di Bucchianico, A. & Einmahl, J.H.J. & Mushkudiani, N.A., 2001. "Smallest nonparametric tolerance regions," Other publications TiSEM 436f9be2-d0ad-49af-b6df-9, Tilburg University, School of Economics and Management.
    16. Elena Di Bernardino & Thomas Laloë & Véronique Maume-Deschamps & Clémentine Prieur, 2013. "Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory," Post-Print hal-00580624, HAL.
    17. Polonik, Wolfgang & Yao, Qiwei, 2002. "Set-Indexed Conditional Empirical and Quantile Processes Based on Dependent Data," Journal of Multivariate Analysis, Elsevier, vol. 80(2), pages 234-255, February.

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