Abstract
In this paper, we propose a nonparametric estimator of the cumulative quantile regression (CQR) function when the response is subjected to random truncation and censorship. The observed responses are weighted by the increments of the product-limit estimator for the underlying response distribution. Strong Gaussian approximations of the associated weighted partial sum process and the CQR process are established under appropriate assumptions. A functional law of the iterated logarithm for the CQR process is also derived. The construct provides a foundation for the asymptotic theory of functional statistics based on these processes.
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References
M. Csörgő and P. Révész, Strong Approximations in Probability and Statistics. (Academic Press, New York, 1981).
M. Csörgő, Quantile Processes with Statistical Applications (SIAM, Philadelphia, 1983).
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.
Yu. Davydov and V. Egorov, “Functional Limit Theorems for Induced Order Statistics”, Math. Methods Statist. 9, 297–313 (2000).
E. Furman and R. Zitikis, “Weighted Risk Capital Allocations”, Insurance: Mathematics and Economics 43, 263–269 (2008).
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).
J. L. Gastwirth, ”A General Definition of the Lorenz Curve”, Econometrica 39, 1037–1039 (1971).
J. L. Gastwirth, “The Estimation of the Lorenz Curve and Gini Index”, Rev. Econometric Statist. 54, 306–316 (1972).
C. M. Goldie, “Convergence Theorems for Empirical Lorenz Curve and Their Inverses”, Adv. Appl. Probab. 9, 765–791 (1977).
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).
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).
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).
C. R. Rao and L. C. Zhao, “Convergence Theorems for Empirical Cumulative Quantile Regression Functions”, Math. Methods Statist. 4, 81–91 and 359 (1995).
C. R. Rao and L. C. Zhao, “Law of the Iterated Logarithm for Empirical Cumulative Quantile Regression Functions”, Statistica Sinica 6, 693–702 (1996).
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).
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).
S. M. Tse, “Strong Gaussian Approximations in the Left Truncated and Right Censored Model”, Statistica Sinica 13, 275–282 (2003).
S. M. Tse, “Quantile Processes for Left Truncated and Right Censored Data”, Ann. Inst. Statist. Math. 57, 61–69 (2005).
S. M. Tse, “On the Empirical Cumulative Quantile Regression Process”, Math. Methods Statist. 18(3), 270–279 (2009). Also available on SpringerLink DOI:10.3103/S1066530709030053.
S. M. Tse, “The Cumulative Quantile Regression Function with Censored and Truncated Covariate”, Math. Methods Statist. 21(3), 238–249 (2012). Also available on SpringerLink DOI:10.3103/S1066530712030040.
Y. Zhou and S. F. Yip, “A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data”, J. Multivar. Anal. 69, 261–280 (1999).
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Tse, S.M. The cumulative quantile regression function with censored and truncated response. Math. Meth. Stat. 24, 147–155 (2015). https://doi.org/10.3103/S1066530715020052
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DOI: https://doi.org/10.3103/S1066530715020052
Keywords
- quantile regression function
- strong Gaussian approximation
- induced order statistics
- product-limit estimator
- weighted partial sum process
- law of the iterated logarithm