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Uniform Representation of Product‐Limit Integrals with Applications

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  • CÉSAR SÁNCHEZ SELLERO
  • WENCESLAO GONZÁLEZ MANTEIGA
  • INGRID VAN KEILEGOM

Abstract

. Let X be a d‐variate random vector that is completely observed, and let Y be a random variable that is subject to right censoring and left truncation. For arbitrary functions ϕ we consider expectations of the form E[ϕ(X, Y)], which appear in many statistical problems, and we estimate these expectations by using a product‐limit estimator for censored and truncated data, extended to the context where covariates are present. An almost sure representation for these estimators is obtained, with a remainder term that is of a certain negligible order, uniformly over a class of ϕ‐functions. This uniformity is important for the application to goodness‐of‐fit testing in regression and to inference for the regression depth, which we consider in more detail.

Suggested Citation

  • César Sánchez Sellero & Wenceslao González Manteiga & Ingrid Van Keilegom, 2005. "Uniform Representation of Product‐Limit Integrals with Applications," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 563-581, December.
    • Handle: RePEc:bla:scjsta:v:32:y:2005:i:4:p:563-581
    DOI: 10.1111/j.1467-9469.2005.00453.x
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    Cited by:

    1. Mercedes Conde‐Amboage & Ingrid Van Keilegom & Wenceslao González‐Manteiga, 2021. "A new lack‐of‐fit test for quantile regression with censored data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 655-688, June.
    2. Gribkova, Svetlana, 2015. "Vector quantization and clustering in the presence of censoring," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 220-233.
    3. Lopez, Olivier, 2019. "A censored copula model for micro-level claim reserving," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 1-14.
    4. Candida Geerdens & Paul Janssen & Ingrid Van Keilegom, 2020. "Goodness-of-fit test for a parametric survival function with cure fraction," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(3), pages 768-792, September.
    5. Wenceslao González-Manteiga & Rosa Crujeiras, 2013. "An updated review of Goodness-of-Fit tests for regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 361-411, September.
    6. Guibert, Quentin & Planchet, Frédéric, 2018. "Non-parametric inference of transition probabilities based on Aalen–Johansen integral estimators for acyclic multi-state models: application to LTC insurance," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 21-36.
    7. Cao, Ricardo & Gonzalez-Manteiga, Wenceslao, 2008. "Goodness-of-fit tests for conditional models under censoring and truncation," Journal of Econometrics, Elsevier, vol. 143(1), pages 166-190, March.
    8. Olivier Lopez, 2007. "On the Estimation of the Joint Distribution in Regression Models with Censored Responses," Working Papers 2007-11, Center for Research in Economics and Statistics.
    9. Lopez, O. & Patilea, V., 2009. "Nonparametric lack-of-fit tests for parametric mean-regression models with censored data," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 210-230, January.
    10. Jacobo Uña-Álvarez & Noël Veraverbeke, 2013. "Generalized copula-graphic estimator," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 343-360, June.
    11. Olivier Lopez & Xavier Milhaud & Pierre-Emmanuel Thérond, 2016. "Tree-based censored regression with applications in insurance," Post-Print hal-01141228, HAL.
    12. García, A., 2016. "Oaxaca-Blinder Type Counterfactual Decomposition Methods for Duration Outcomes," Documentos de Trabajo 14186, Universidad del Rosario.
    13. Olivier Lopez & Valentin Patilea, 2007. "Nonparametric Lack-of-fit Tests for Parametric Mean-Regression Model with Censored Data," Working Papers 2007-01, Center for Research in Economics and Statistics.
    14. Shi, Jianhua & Chen, Xiaoping & Zhou, Yong, 2015. "The strong representation for the nonparametric estimator of length-biased and right-censored data," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 49-57.
    15. Lopez, Olivier, 2012. "A generalization of the Kaplan–Meier estimator for analyzing bivariate mortality under right-censoring and left-truncation with applications in model-checking for survival copula models," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 505-516.
    16. Jacobo de Uña‐Álvarez & Micha Mandel, 2018. "Nonparametric estimation of transition probabilities for a general progressive multi‐state model under cross‐sectional sampling," Biometrics, The International Biometric Society, vol. 74(4), pages 1203-1212, December.
    17. Jacobo Uña-Álvarez, 2013. "Comments on: An updated review of Goodness-of-Fit tests for regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 414-418, September.

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