Asymptotic properties of conditional U-statistics using delta sequences

References

• Abrevaya, J., and W. Jiang. 2005a. A nonparametric approach to measuring and testing curvature. Journal of Business & Economic Statistics 23 (1):1–19. doi: 10.1198/073500104000000316.
   Web of Science ®Google Scholar
• Ahmad, I. A. 1981. A note on nonparametric density estimation for dependent variables using a delta sequence. Annals of the Institute of Statistical Mathematics 33 (2):247–54. doi: 10.1007/BF02480938.
   Web of Science ®Google Scholar
• Arcones, M. A., and E. Giné. 1993. Limit theorems for U-processes. The Annals of Probability 21 (3):1494–542. doi: 10.1214/aop/1176989128.
   Web of Science ®Google Scholar
• Arcones, M. A., and E. Giné. 1995. On the law of the iterated logarithm for canonical U-statistics and processes. Stochastic Processes and Their Applications 58 (2):217–45. doi: 10.1016/0304-4149(94)00023-M.
   Web of Science ®Google Scholar
• Arcones, M. A., and Y. Wang. 2006. Some new tests for normality based on U-processes. Statistics & Probability Letters 76 (1):69–82. doi: 10.1016/j.spl.2005.07.003.
   Web of Science ®Google Scholar
• Arcones, M. A., Z. Chen, and E. Giné. 1994. Estimators related to U-processes with applications to multivariate medians: Asymptotic normality. The Annals of Statistics 22 (3):1460–77. doi: 10.1214/aos/1176325637.
   Web of Science ®Google Scholar
• Basu, A. K., and D. K. Sahoo. 1985. Speed of convergence in density estimation by delta sequence and estimation of mode. Calcutta Statistical Association Bulletin 34 (1-2):1–13. doi: 10.1177/0008068319850101.
   Google Scholar
• Belitser, E. 2000. Local minimax pointwise estimation of a multivariate density. Statistica Neerlandica 54 (3):351–65. doi: 10.1111/1467-9574.00145].
   Web of Science ®Google Scholar
• Bellet, A., and A. Habrard. 2015. Robustness and generalization for metric learning. Neurocomputing 151:259–67. doi: 10.1016/j.neucom.2014.09.044.
   Web of Science ®Google Scholar
• Bellet, A., A. Habrard, and M. Sebban. 2013. A survey on metric learning for feature vectors and structured data. CoRR, Abs/1306.6709.
   Google Scholar
• Bergsma, W., and A. Dassios. 2014. A consistent test of independence based on a sign covariance related to Kendall’s tau. Bernoulli 20 (2):1006–28. doi: 10.3150/13-BEJ514.
   Web of Science ®Google Scholar
• Blum, J. R., J. Kiefer, and M. Rosenblatt. 1961. Distribution free tests of independence based on the sample distribution function. The Annals of Mathematical Statistics 32 (2):485–98. doi: 10.1214/aoms/1177705055.
   Google Scholar
• Blundell, R., A. Gosling, H. Ichimura, and C. Meghir. 2007. Changes in the distribution of male and female wages accounting for employment composition using bounds. Econometrica 75 (2):323–63. doi: 10.1111/j.1468-0262.2006.00750.x.
   Web of Science ®Google Scholar
• Borovkova, S., R. Burton, and H. Dehling. 1999. Consistency of the Takens estimator for the correlation dimension. The Annals of Applied Probability 9 (2):376–90. doi: 10.1214/aoap/1029962747.
   Web of Science ®Google Scholar
• Borovkova, S., R. Burton, and H. Dehling. 2001. Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation. Transactions of the American Mathematical Society 353 (11):4261–318. doi: 10.1090/S0002-9947-01-02819-7.
   Web of Science ®Google Scholar
• Borovskikh, Y. V. 1996. U -statistics in Banach spaces. Utrecht: VSP.
   Google Scholar
• Bouzebda, S. 2011. Some new multivariate tests of independence. Mathematical Methods of Statistics 20 (3):192–205. doi: 10.3103/S1066530711030021.
   Google Scholar
• Bouzebda, S. 2012. On the strong approximation of bootstrapped empirical copula processes with applications. Mathematical Methods of Statistics 21 (3):153–88. doi: 10.3103/S1066530712030015.
   Google Scholar
• Bouzebda, S. 2014. General tests of independence based on empirical processes indexed by functions. Statistical Methodology 21:59–87. doi: 10.1016/j.stamet.2014.03.001.
   Google Scholar
• Bouzebda, S. 2023. General tests of conditional independence based on empirical processes indexed by functions. Japanese Journal of Statistics and Data Science 1–65. doi: 10.1080/03610926.2023.2179887.
   Google Scholar
• Bouzebda, S., and M. Chaouch. 2022. Uniform limit theorems for a class of conditional Z-estimators when covariates are functions. Journal of Multivariate Analysis. 189: 104872, 21. doi: 10.1016/j.jmva.2021.104872.
   Web of Science ®Google Scholar
• Bouzebda, S., and S. Didi. 2017a. Additive regression model for stationary and ergodic continuous time processes. Communications in Statistics - Theory and Methods 46 (5):2454–93. doi: 10.1080/03610926.2015.1048882.
   Web of Science ®Google Scholar
• Bouzebda, S., and S. Didi. 2017b. Multivariate wavelet density and regression estimators for stationary and ergodic discrete time processes: Asymptotic results. Communications in Statistics - Theory and Methods 46 (3):1367–406. doi: 10.1080/03610926.2015.1019144.
   Web of Science ®Google Scholar
• Bouzebda, S., and S. Didi. 2021. Some asymptotic properties of kernel regression estimators of the mode for stationary and ergodic continuous time processes. Revista Matematica Complutense 34 (3):811–52. doi: 10.1007/s13163-020-00368-6.
   PubMed Web of Science ®Google Scholar
• Bouzebda, S., and S. Didi. 2022. Some results about kernel estimators for function derivatives based on stationary and ergodic continuous time processes with applications. Communications in Statistics - Theory and Methods 51 (12):3886–933. doi: 10.1080/03610926.2020.1805466.
   Web of Science ®Google Scholar
• Bouzebda, S., and T. El-Hadjali. 2020. Uniform convergence rate of the kernel regression estimator adaptive to intrinsic dimension in presence of censored data. Journal of Nonparametric Statistics 32 (4):864–914. doi: 10.1080/10485252.2020.1834107.
   Web of Science ®Google Scholar
• Bouzebda, S., and A. Keziou. 2010a. New estimates and tests of independence in semiparametric copula models. Kybernetika (Prague) 46 (1):178–201.
   Web of Science ®Google Scholar
• Bouzebda, S., and A. Keziou. 2010b. A new test procedure of independence in copula models via χ2 -divergence. Communications in Statistics - Theory and Methods 39 (1-2):1–20. doi: 10.1080/03610920802645379.
   Web of Science ®Google Scholar
• Bouzebda, S., and B. Nemouchi. 2019. Central limit theorems for conditional empirical and conditional U-processes of stationary mixing sequences. Mathematical Methods of Statistics 28 (3):169–207. doi: 10.3103/S1066530719030013.
   Web of Science ®Google Scholar
• Bouzebda, S., and B. Nemouchi. 2020. Uniform consistency and uniform in bandwidth consistency for nonparametric regression estimates and conditional U-statistics involving functional data. Journal of Nonparametric Statistics 32 (2):452–509. doi: 10.1080/10485252.2020.1759597.
   Web of Science ®Google Scholar
• Bouzebda, S. and Nemouchi, B. 2022. Weak-convergence of empirical conditional processes and conditional U-processes involving functional mixing data. Statistical Inference for Stochastic Processes. To appear 1–56. doi: 10.1007/s11203-022-09276-6.
   Google Scholar
• Bouzebda, S., and A. Nezzal. 2022. Uniform consistency and uniform in number of neighbors consistency for nonparametric regression estimates and conditional U-statistics involving functional data. Japanese Journal of Statistics and Data Science 5 (2):431–533. doi: 10.1007/s42081-022-00161-3.
   Google Scholar
• Bouzebda, S., and I. Soukarieh. 2023a. Non-parametric conditional U-processes for locally stationary functional random fields under stochastic sampling design. Mathematics 11 (1):1–70. doi: 10.3390/math11010016.
   Web of Science ®Google Scholar
• Bouzebda, S., and Soukarieh, I. 2022. Renewal type bootstrap for U-process Markov chains. Markov Processes and Related Fields 28 (5):673–735.
   Web of Science ®Google Scholar
• Bouzebda, S., and N. Taachouche. 2023. Rates of the strong uniform consistency for the kernel-type regression function estimators with general kernels on manifolds. Mathematical Methods of Statistics 23 (1):1–62.
   Google Scholar
• Bouzebda, S., and T. Zari. 2013. Asymptotic behavior of weighted multivariate Cramér-von Mises-type statistics under contiguous alternatives. Mathematical Methods of Statistics 22 (3):226–52. doi: 10.3103/S1066530713030046.
   Google Scholar
• Bouzebda, S., Chaouch, M. D, and Biha, S. 2022. Asymptotics for function derivatives estimators based on stationary and ergodic discrete time processes. Annals of the Institute of Statistical Mathematics 74 (4):737–71. doi: 10.1007/s10463-021-00814-2.
   Web of Science ®Google Scholar
• Bouzebda, S., T. El-hadjali, and A. A. Ferfache 2022. Uniform in bandwidth consistency of conditional U-statistics adaptive to intrinsic dimension in presence of censored data. Sankhya A 1–59. doi: 10.1007/s13171-022-00301-7.
   PubMedGoogle Scholar
• Bouzebda, S., S. Didi, and L. El Hajj. 2015. Multivariate wavelet density and regression estimators for stationary and ergodic continuous time processes: Asymptotic results. Mathematical Methods of Statistics 24 (3):163–99. doi: 10.3103/S1066530715030011.
   Google Scholar
• Bouzebda, S., I. Elhattab, and B. Nemouchi. 2021. On the uniform-in-bandwidth consistency of the general conditional U-statistics based on the copula representation. Journal of Nonparametric Statistics 33 (2):321–58. doi: 10.1080/10485252.2021.1937621.
   Web of Science ®Google Scholar
• Bouzebda, S., A. Nezzal, and T. Zari. 2023. Uniform consistency for functional conditional U-statistics using delta-sequences. Mathematics 11 (1):1–39.
   Web of Science ®Google Scholar
• Brunel, E., and F. Comte. 2006. Adaptive nonparametric regression estimation in presence of right censoring. Mathematical Methods of Statistics 15 (3):233–55.
   Google Scholar
• Cao, Q., Z.-C. Guo, and Y. Ying. 2016. Generalization bounds for metric and similarity learning. Machine Learning 102 (1):115–32. doi: 10.1007/s10994-015-5499-7.
   Web of Science ®Google Scholar
• Carbonez, A., L. Györfi, and E. C. van der Meulen. 1995. Partitioning-estimates of a regression function under random censoring. Statistics and Decisions 13 (1):21–37.
   Google Scholar
• Chen, X., and K. Kato. 2020. Jackknife multiplier bootstrap: Finite sample approximations to the U-process supremum with applications. Probability Theory and Related Fields 176 (3-4):1097–163. doi: 10.1007/s00440-019-00936-y.
   Web of Science ®Google Scholar
• Chen, Y., and S. Datta. 2019. Adjustments of multi-sample U-statistics to right censored data and confounding covariates. Computational Statistics & Data Analysis 135:1–14. doi: 10.1016/j.csda.2019.01.012.
   Web of Science ®Google Scholar
• Chentsov, N. N. 1962. Evaluation of an unknown distribution density from observations. Soviet Mathematics - Doklady 3:1559–62.
   Google Scholar
• Clémençon, S., I. Colin, and A. Bellet. 2016. Scaling-up empirical risk minimization: Optimization of incomplete U-statistics. Journal of Machine Learning Research 17 (76):36.
   Google Scholar
• Clémençon, S., G. Lugosi, and N. Vayatis. 2008. Ranking and empirical minimization of U-statistics. The Annals of Statistics 36 (2):844–74. doi: 10.1214/009052607000000910.
   Web of Science ®Google Scholar
• Clémençon, S., and S. Robbiano. 2015. The TreeRank tournament algorithm for multipartite ranking. Journal of Nonparametric Statistics 27 (1):107–26. doi: 10.1080/10485252.2014.967770.
   Web of Science ®Google Scholar
• Clémençon, S., S. Robbiano, and N. Vayatis. 2013. Ranking data with ordinal labels: Optimality and pairwise aggregation. Machine Learning 91 (1):67–104. doi: 10.1007/s10994-012-5325-4.
   Web of Science ®Google Scholar
• Cristóbal Cristóbal, J. A., Faraldo Roca, P. G, and Manteiga, W. 1987. A class of linear regression parameter estimators constructed by nonparametric estimation. The Annals of Statistics 15 (2):603–9. doi: 10.1214/aos/1176350363.
   Web of Science ®Google Scholar
• Datta, S., D. Bandyopadhyay, and G. A. Satten. 2010. Inverse probability of censoring weighted U-statistics for right-censored data with an application to testing hypotheses. Scandinavian Journal of Statistics 37 (4):680–700. doi: 10.1111/j.1467-9469.2010.00697.x.
   Web of Science ®Google Scholar
• de la Peña, V. H., and E. Giné. 1999. From dependence to independence, Randomly stopped processes. U-statistics and processes. Martingales and beyond. In Decoupling. Probability and its Applications (New York). New York: Springer-Verlag,
   Google Scholar
• Deheuvels, P. 2011. One bootstrap suffices to generate sharp uniform bounds in functional estimation. Kybernetika (Prague) 47 (6):855–65.
   Web of Science ®Google Scholar
• Denker, M., and G. Keller. 1983. On U-statistics and v. Mises’ statistics for weakly dependent processes. Zeitschrift fürWahrscheinlichkeitstheorie Und Verwandte Gebiete 64 (4):505–22. doi: 10.1007/BF00534953.
   Web of Science ®Google Scholar
• Derumigny, A. 2019. Estimation of a regular conditional functional by conditional U-statistics regression. arXiv preprint arXiv:1903.10914.
   Google Scholar
• Dony, J., and D. M. Mason. 2008. Uniform in bandwidth consistency of conditional U-statistics. Bernoulli 14 (4):1108–33. doi: 10.3150/08-BEJ136.
   Web of Science ®Google Scholar
• Dwass, M. 1956. The large-sample power of rank order tests in the two-sample problem. The Annals of Mathematical Statistics 27 (2):352–74. doi: 10.1214/aoms/1177728263.
   Google Scholar
• Ellison, G., and S. F. Ellison. 2011. Strategic entry deterrence and the behavior of pharmaceutical incumbents prior to patent expiration. American Economic Journal: Microeconomics 3 (1):1–36. doi: 10.1257/mic.3.1.1.
   Web of Science ®Google Scholar
• Fan, J., and I. Gijbels. 1996. Local polynomial modelling and its applications. In Volume 66 of Monographs on Statistics and Applied Probability. London: Chapman & Hall.
   Google Scholar
• Földes, A., and L. Rejtő. 1981. A LIL type result for the product limit estimator. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 56 (1):75–86. doi: 10.1007/BF00531975.
   Web of Science ®Google Scholar
• Földes, A., and P. Révész. 1974. A general method for density estimation. Studia Scientiarum Mathematicarum Hungarica 9:81–92.
   Google Scholar
• Gao, J., and I. Gijbels. 2008. Bandwidth selection in nonparametric kernel testing. Journal of the American Statistical Association 103 (484):1584–94. doi: 10.1198/016214508000000968.
   Web of Science ®Google Scholar
• Ghosal, S., A. Sen, and A. W. van der Vaart. 2000. Testing monotonicity of regression. The Annals of Statistics 28 (4):1054–82. doi: 10.1214/aos/1015956707.
   Web of Science ®Google Scholar
• Giné, E., and D. M. Mason. 2007a. Laws of the iterated logarithm for the local U-statistic process. Journal of Theoretical Probability 20 (3):457–85. doi: 10.1007/s10959-007-0067-0.
   Web of Science ®Google Scholar
• Giné, E., and D. M. Mason. 2007b. On local U-statistic processes and the estimation of densities of functions of several sample variables. The Annals of Statistics 35 (3):1105–45. doi: 10.1214/009053607000000154.
   Web of Science ®Google Scholar
• Györfi, L., M. Kohler, A. Krzyżak, and H. Walk. 2002. A distribution-free theory of nonparametric regression. In Springer Series in Statistics. New York: Springer-Verlag.
   Google Scholar
• Halmos, P. R. 1946. The theory of unbiased estimation. The Annals of Mathematical Statistics 17 (1):34–43. doi: 10.1214/aoms/1177731020.
   Google Scholar
• Harel, M., and M. L. Puri. 1996. Conditional U-statistics for dependent random variables. Journal of Multivariate Analysis. 57 (1):84–100. doi: 10.1006/jmva.1996.0023.
   Web of Science ®Google Scholar
• Hengartner, N. W., and S. Sperlich. 2005. Rate optimal estimation with the integration method in the presence of many covariates. Journal of Multivariate Analysis. 95 (2):246–72. doi: 10.1016/j.jmva.2004.09.010.
   Web of Science ®Google Scholar
• Hoeffding, W. 1948. A class of statistics with asymptotically normal distribution. The Annals of Mathematical Statistics 19 (3):293–325. doi: 10.1214/aoms/1177730196.
   Google Scholar
• Hoeffding, W. 1963. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58 (301):13–30. doi: 10.1080/01621459.1963.10500830.
   Web of Science ®Google Scholar
• Hollander, M., and F. Proschan. 1972. Testing whether new is better than used. The Annals of Mathematical Statistics 43 (4):1136–46. doi: 10.1214/aoms/1177692466.
   Google Scholar
• Horowitz, J. L., and V. G. Spokoiny. 2001. An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69 (3):599–631. doi: 10.1111/1468-0262.00207.
   Web of Science ®Google Scholar
• Isogai, E. 1990. Nonparametric estimation of a regression function by delta sequences. Annals of the Institute of Statistical Mathematics 42 (4):699–708. doi: 10.1007/BF02481145.
   Web of Science ®Google Scholar
• Jin, R., S. Wang, and Y. Zhou. 2009. Regularized distance metric learning: Theory and algorithm. In Advances in neural information processing systems, eds. Y. Bengio, D. Schuurmans, J. Lafferty, C. Williams, and A. Culotta, vol 22. Proceedings of a meeting held 7-10 December 2009, Vancouver, British Columbia, Canada. Curran Associates, Inc.
   Google Scholar
• Joly, E., and G. Lugosi. 2016. Robust estimation of U-statistics. Stochastic Processes and Their Applications 126 (12):3760–73. doi: 10.1016/j.spa.2016.04.021.
   Web of Science ®Google Scholar
• Jones, M. C., S. J. Davies, and B. U. Park. 1994. Versions of kernel-type regression estimators. Journal of the American Statistical Association 89 (427):825–32. doi: 10.1080/01621459.1994.10476816.
   Web of Science ®Google Scholar
• Kaplan, E. L., and P. Meier. 1958. Nonparametric estimation from incomplete observations. Journal of the American Statistical Association 53 (282):457–81. doi: 10.1080/01621459.1958.10501452.
   Web of Science ®Google Scholar
• Kendall, M. G. 1938. A new measure of rank correlation. Biometrika 30 (1-2):81–93. doi: 10.1093/biomet/30.1-2.81.
   Google Scholar
• Kohler, M., K. Máthé, and M. Pintér. 2002. Prediction from randomly right censored data. Journal of Multivariate Analysis. 80 (1):73–100. doi: 10.1006/jmva.2000.1973.
   Web of Science ®Google Scholar
• Koroljuk, V. S., and Y. V. Borovskich. 1994. Theory of U-statistics, volume 273 of mathematics and its applications. Dordrecht: Kluwer Academic Publishers Group (Translated from the 1989 Russian original by P. V. Malyshev and D. V. Malyshev and revised by the authors).
   Google Scholar
• Lee, A. J. 1990. Theory and practice. In U -statistics, volume 110 of statistics: Textbooks and monographs. New York: Marcel Dekker Inc.
   Google Scholar
• Lee, S., O. Linton, and Y.-J. Whang. 2009. Testing for stochastic monotonicity. Econometrica 77 (2):585–602.
   Web of Science ®Google Scholar
• Lehmann, E. L. 1951. A general concept of unbiasedness. The Annals of Mathematical Statistics 22 (4):587–92. doi: 10.1214/aoms/1177729549.
   Google Scholar
• Leucht, A. 2012. Degenerate U- and V-statistics under weak dependence: Asymptotic theory and bootstrap consistency. Bernoulli 18 (2):552–85. doi: 10.3150/11-BEJ354.
   Web of Science ®Google Scholar
• Leucht, A., and M. H. Neumann. 2013. Degenerate U- and V-statistics under ergodicity: Asymptotics, bootstrap and applications in statistics. Annals of the Institute of Statistical Mathematics 65 (2):349–86. doi: 10.1007/s10463-012-0374-9.
   Web of Science ®Google Scholar
• Linton, O. B., and D. T. Jacho-Chávez. 2010. On internally corrected and symmetrized kernel estimators for nonparametric regression. TEST 19 (1):166–86. doi: 10.1007/s11749-009-0145-y.
   Web of Science ®Google Scholar
• Mack, Y. P., and H.-G. Müller. 1989. Derivative estimation in nonparametric regression with random predictor variable. Sankhyā Series A 51 (1):59–72.
   Google Scholar
• Maillot, B., and V. Viallon. 2009. Uniform limit laws of the logarithm for nonparametric estimators of the regression function in presence of censored data. Mathematical Methods of Statistics 18 (2):159–84. doi: 10.3103/S1066530709020045.
   Google Scholar
• Marron, J. S., and W. Härdle. 1986. Random approximations to some measures of accuracy in nonparametric curve estimation. Journal of Multivariate Analysis. 20 (1):91–113. doi: 10.1016/0047-259X(86)90021-7.
   Web of Science ®Google Scholar
• Nadaraja, E. A. 1964. On a regression estimate. Akademija Nauk SSSR. Teorija Verojatnostei i ee Primenenija 9:157–9.
   Google Scholar
• Nadaraja, E. A. 1965. On non-parametric estimates of density functions and regression. Akademija Nauk SSSR. Teorija Verojatnostei i ee Primenenija 10:199–203.
   Google Scholar
• Nolan, D., and J. S. Marron. 1989. Uniform consistency of automatic and location-adaptive delta-sequence estimators. Probability Theory and Related Fields 80 (4):619–32. doi: 10.1007/BF00318909.
   Web of Science ®Google Scholar
• Ouassou, I., and M. Rachdi. 2012. Regression operator estimation by delta-sequences method for functional data and its applications. AStA Advances in Statistical Analysis 96 (4):451–65. doi: 10.1007/s10182-011-0175-0.
   Web of Science ®Google Scholar
• Parzen, E. 1962. On estimation of a probability density function and mode. The Annals of Mathematical Statistics 33 (3):1065–76. doi: 10.1214/aoms/1177704472.
   Google Scholar
• Piterbarg, V. I. 1996. Asymptotic methods in the theory of Gaussian processes and fields, volume 148 of Translations of Mathematical Monographs. Providence, RI: American Mathematical Society (Translated from the Russian by V. V. Piterbarg, Revised by the author).
   Google Scholar
• Rao, B. L. S. P. 1978. Density estimation for Markov processes using delta-sequences. Annals of the Institute of Statistical Mathematics 30 (2):321–8.
   Web of Science ®Google Scholar
• Rao, B. L. S. P. 2010. Nonparametric density estimation for functional data by delta sequences. Brazilian Journal of Probability and Statistics 24 (3):468–78.
   Web of Science ®Google Scholar
• Rao, B. L. S. P., and A. Sen. 1995. Limit distributions of conditional U-statistics. Journal of Theoretical Probability 8 (2):261–301. doi: 10.1007/BF02212880.
   Web of Science ®Google Scholar
• Rejchel, W. 2012. On ranking and generalization bounds. Journal of Machine Learning Research 13:1373–92.
   Web of Science ®Google Scholar
• Schick, A., Y. Wang, and W. Wefelmeyer. 2011. Tests for normality based on density estimators of convolutions. Statistics & Probability Letters 81 (2):337–43. doi: 10.1016/j.spl.2010.10.022.
   Web of Science ®Google Scholar
• Sen, A. 1994. Uniform strong consistency rates for conditional U-statistics. Sankhyā Series A 56 (2):179–94.
   Google Scholar
• Serfling, R. J. 1980. Approximation theorems of mathematical statistics. In Wiley Series in Probability and Mathematical Statistics. New York: John Wiley & Sons, Inc.
   Google Scholar
• Shang, H. L. 2014. Bayesian bandwidth estimation for a functional nonparametric regression model with mixed types of regressors and unknown error density. Journal of Nonparametric Statistics 26 (3):599–615. doi: 10.1080/10485252.2014.916806.
   Web of Science ®Google Scholar
• Shen, J., and Y. Xie. 2013. Strong consistency of the internal estimator of nonparametric regression with dependent data. Statistics & Probability Letters 83 (8):1915–25. doi: 10.1016/j.spl.2013.04.027.
   Web of Science ®Google Scholar
• Sherman, R. P. 1993. The limiting distribution of the maximum rank correlation estimator. Econometrica 61 (1):123–37. doi: 10.2307/2951780.
   Web of Science ®Google Scholar
• Sherman, R. P. 1994. Maximal inequalities for degenerate U-processes with applications to optimization estimators. The Annals of Statistics 22 (1):439–59. doi: 10.1214/aos/1176325377.
   Web of Science ®Google Scholar
• Silverman, B. W. 1978. Distances on circles, toruses and spheres. Journal of Applied Probability 15 (1):136–43. doi: 10.2307/3213243.
   Web of Science ®Google Scholar
• Silverman, B. W. 1986. Density estimation for statistics and data analysis. Monographs on Statistics and Applied Probability. London: Chapman & Hall.
   Google Scholar
• Solon, G. 1992. Intergenerational income mobility in the united states. The American Economic Review 82 (3):393–408.
   Web of Science ®Google Scholar
• Soukarieh, I., and S. Bouzebda. 2022. Exchangeably weighted bootstraps of general Markov U-process. Mathematics 10 (20):3745–2. doi: 10.3390/math10203745.
   Web of Science ®Google Scholar
• Soukarieh, I., and S. Bouzebda. 2023. Renewal type bootstrap for increasing degree U-process of a Markov chain. Journal of Multivariate Analysis. 195:105143. doi: 10.1016/j.jmva.2022.105143.
   Web of Science ®Google Scholar
• Stute, W. 1991. Conditional U-statistics. The Annals of Probability 19 (2):812–25. doi: 10.1214/aop/1176990452.
   Web of Science ®Google Scholar
• Stute, W. 1993. Almost sure representations of the product-limit estimator for truncated data. The Annals of Statistics 21 (1):146–56. doi: 10.1214/aos/1176349019.
   Web of Science ®Google Scholar
• Stute, W. 1994a. Lp-convergence of conditional U-statistics. Journal of Multivariate Analysis 51 (1): 71–82. doi: 10.1006/jmva.1994.1050.
   Web of Science ®Google Scholar
• Stute, W. 1994b. Universally consistent conditional U-statistics. The Annals of Statistics 22 (1):460–73. doi: 10.1214/aos/1176325378.
   Web of Science ®Google Scholar
• Stute, W. 1996. Symmetrized NN-conditional U-statistics. In Research developments in probability and statistics, 231–7. Utrecht: VSP.
   Google Scholar
• Stute, W., and J.-L. Wang. 1993. Multi-sample U-statistics for censored data. Scandinavian Journal of Statistics 20 (4):369–74.
   Web of Science ®Google Scholar
• Susarla, V., and G. Walter. 1981. Estimation of a multivariate density function using delta sequences. The Annals of Statistics 9 (2):347–55. doi: 10.1214/aos/1176345400.
   Web of Science ®Google Scholar
• von Mises, R. 1947. On the asymptotic distribution of differentiable statistical functions. Annals of Mathematical Statistics 18:309–48.
   Google Scholar
• Walter, G. G. 1983. A general approach to classification problems. Information Sciences 30 (1):67–77. doi: 10.1016/0020-0255(83)90047-6.
   Web of Science ®Google Scholar
• Walter, G., and J. Blum. 1979. Probability density estimation using delta sequences. The Annals of Statistics 7 (2):328–40. doi: 10.1214/aos/1176344617.
   Web of Science ®Google Scholar
• Wand, M. P., and M. C. Jones. 1995. Kernel smoothing. In volume 60 of Monographs on Statistics and Applied Probability. London: Chapman and Hall, Ltd.
   Google Scholar
• Watson, G. S. 1964. Smooth regression analysis. Sankhyā Series A 26:359–72.
   Google Scholar
• Watson, G. S., and M. R. Leadbetter. 1964. Hazard analysis. II. Sankhyā Series A 26:101–16.
   Google Scholar
• Winter, B. B. 1975. Rate of strong consistency of two nonparametric density estimators. The Annals of Statistics 3 (3):759–66. doi: 10.1214/aos/1176343142.
   Web of Science ®Google Scholar
• Yuan, A., M. Giurcanu, G. Luta, and M. T. Tan. 2017. U-statistics with conditional kernels for incomplete data models. Annals of the Institute of Statistical Mathematics 69 (2):271–302. doi: 10.1007/s10463-015-0537-6.
   Web of Science ®Google Scholar