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The cumulative quantile regression function with censored and truncated covariate

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Abstract

In this paper, we consider the empirical estimator of the cumulative quantile regression (CQR) functionwhen the covariate is subjected to random truncation and censorship. Strong Gaussian approximations for the associated CQR process are established under appropriate assumptions. A functional law of the iterated logarithm for the CQR process is also derived. These results provide a foundation for the asymptotic theory of functional statistics based on these processes.

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References

  1. M. Csörgő and P. Révész, Strong Approximations in Probability and Statistics (Academic Press, New York, 1981).

    Google Scholar 

  2. M. Csörgő, Quantile Processes with Statistical Applications (SIAM, Philadelphia, 1983).

    Book  Google Scholar 

  3. M. Csörgő, S. Csörgő, and L. Horváth, An Asymptotic Theory for Empirical Reliability and Concentration Processes, in Lecture Notes in Statistics (Springer, New York, 1986), Vol. 33.

    Google Scholar 

  4. Yu. Davydov and V. Egorov, “Functional Limit Theorems for Induced Order Statistics”, Math. Methods Statist. 9, 297–313 (2000).

    MathSciNet  MATH  Google Scholar 

  5. E. Furman and R. Zitikis, “Weighted Risk Capital Allocations”, Insurance:Mathematics and Economics 43, 263–269 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  6. E. Furman and R. Zitikis, “A Monotonicity Property of the Composition of Regularized and Inverted-Regularized Gamma Functions with Applications”, J. Math. Anal. and Appl. 348, 971–976 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  7. J. L. Gastwirth, “A General Definition of the Lorenz Curve”, Econometrica 39, 1037–1039 (1971).

    Article  MATH  Google Scholar 

  8. J. L. Gastwirth, “The Estimation of the Lorenz Curve and Gini Index”, Review of Econometric Statistics 54, 306–316 (1972).

    Article  MathSciNet  Google Scholar 

  9. C. M. Goldie, “Convergence Theorems for Empirical Lorenz Curve and Their Inverses”, Advances in Applied Probab. 9, 765–791 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Greselin, M. L. Puri, and R. Zitikis, “L-functions, Processes, and Statistics in Measuring Economic Inequality and Acturial Risks”, Statist. & Its Interface 2, 227–245 (2009).

    MathSciNet  MATH  Google Scholar 

  11. J. Komlós, P. Major, and G. Tusnády, “An Approximation of Partial Sums of Independent r.v.’s and the Sample d.f. I”, Z.Wahrsch. verw. Gebiete 32, 111–132 (1975).

    Article  MATH  Google Scholar 

  12. J. Komlós, P. Major, and G. Tusnády, “An Approximation of Partial Sums of Independent r.v.’s and the Sample d.f. II”, Z.Wahrsch. verw. Gebiete 34, 33–58 (1976).

    Article  MATH  Google Scholar 

  13. C. R. Rao and L. C. Zhao, “Convergence Theorems for Empirical Cumulative Quantile Regression Functions”, Math. Methods Statist. 4, 81–91 and 359 (1995).

    MathSciNet  MATH  Google Scholar 

  14. C. R. Rao and L. C. Zhao, “Law of the Iterated Logarithm for Empirical Cumulative Quantile Regression Functions”, Statistica Sinica 6, 693–702 (1996).

    MathSciNet  MATH  Google Scholar 

  15. E. Schechtman, A. Shelef, S. Yitzhaki, and R. Zitikis, “Testing Hypotheses about Absolute Concentration Curves and Marginal Conditional Stochastic Dominance”, EconometricTheory 24, 1044–1062 (2008).

    MATH  Google Scholar 

  16. W. Y. Tsai, N. P. Jewel, and M. C. Wang, “A Note on the Product-Limit Estimator under Right Censoring and Left Truncation”, Biometrika 74, 883–886 (1987).

    Article  MATH  Google Scholar 

  17. S. M. Tse, “Strong Gaussian Approximations in the Left Truncated and Right Censored Model”, Statistica Sinica 13, 275–282 (2003).

    MathSciNet  MATH  Google Scholar 

  18. S. M. Tse, “Quantile Processes for Left Truncated and Right Censored Data”, Ann. Inst. Statist. Math. 57, 61–69 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  19. S. M. Tse, “On the Empirical Cumulative Quantile Regression Process”, Math. Methods Statist. 18(3), 270–279 (2009). Also available on Springer Link DOI:10.3103/S1066530709030053.

    Article  MathSciNet  Google Scholar 

  20. Y. Zhou and S. F. Yip, “A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data”, J.Multivariate Analysis 69, 261–280 (1999).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dept. Appl. Math., National Dong Hwa University, Hualien, Taiwan

    S. M. Tse

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  1. S. M. Tse
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Correspondence to S. M. Tse.

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Tse, S.M. The cumulative quantile regression function with censored and truncated covariate. Math. Meth. Stat. 21, 238–249 (2012). https://doi.org/10.3103/S1066530712030040

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  • DOI: https://doi.org/10.3103/S1066530712030040

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