Limit theorems for a class of processes generalizing the U-empirical process

References

• J. Abrevaya and W. Jiang, A nonparametric approach to measuring and testing curvature, J. Bus. Econ. Stat. 23(1) (2005), pp. 1–19.
   Web of Science ®Google Scholar
• D. Aleksić, M. Cuparić, and B. Milošević, Non-degenerate U-statistics for data missing completely at random with application to testing independence, Stat 12 (2023), pp. 14. Paper No. e634.
   Web of Science ®Google Scholar
• M.A. Arcones, Z. Chen, and E. Giné, Estimators related to U-processes with applications to multivariate medians: asymptotic normality, Ann. Stat. 22(3) (1994), pp. 1460–1477.
   Web of Science ®Google Scholar
• M.A. Arcones, The large deviation principle for stochastic processes. I, Teor. Veroyatnost. i Primenen.47(4) (2002), pp. 727–746.
   Web of Science ®Google Scholar
• M.A. Arcones, The large deviation principle for stochastic processes. II, Teor. Veroyatnost. i Primenen.48(1) (2003), pp. 122–150.
   Web of Science ®Google Scholar
• M.A. Arcones and E. Giné, U-processes indexed by Vapnik-Červonenkis classes of functions with applications to asymptotics and bootstrap of U-statistics with estimated parameters, Stoch. Process. Appl. 52(1) (1994), pp. 17–38.
   Web of Science ®Google Scholar
• M.A. Arcones and E. Giné, On the bootstrap of U and V statistics, Ann. Stat. 20(2) (1992), pp. 655–674.
   Web of Science ®Google Scholar
• M.A. Arcones and E. Giné, Limit theorems for U-processes, Ann. Probab. 21(3) (1993), pp. 1494–1542.
   Web of Science ®Google Scholar
• M.A. Arcones and E. Giné, U-processes indexed by Vapnik-červonenkis classes of functions with applications to asymptotics and bootstrap of U-statistics with estimated parameters, Stoch. Process. Appl. 52(1) (1994), pp. 17–38.
   Web of Science ®Google Scholar
• M.A. Arcones and Y. Wang, Some new tests for normality based on U-processes, Stat. Probab. Lett.76(1) (2006), pp. 69–82.
   Web of Science ®Google Scholar
• A. Bellet and A. Habrard, Robustness and generalization for metric learning, Neurocomputing 151 (2015), pp. 259–267.
   Web of Science ®Google Scholar
• A. Bellet, A. Habrard, and M. Sebban, A survey on metric learning for feature vectors and structured data. CoRR, abs/1306.6709, 2013.
   Google Scholar
• D.Z. Bello, M. Valk, and G.B. Cybis, Towards U-statistics clustering inference for multiple groups, J. Stat. Comput. Simul. 94(1) (2024), pp. 204–222.
   Web of Science ®Google Scholar
• W. Bergsma and A. Dassios, A consistent test of independence based on a sign covariance related to Kendall's tau, Bernoulli 20(2) (2014), pp. 1006–1028.
   Web of Science ®Google Scholar
• N. Berrahou, S. Bouzebda, and D. Lahcen, A nonparametric distribution-free test of independence among continuous random vectors based on l1-error, Bernoulli (2024), pp. 1–50.
   Google Scholar
• P. Billingsley, Convergence of Probability Measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc. A Wiley-Interscience Publication, New York, 1999.
   Google Scholar
• J.R. Blum, J. Kiefer, and M. Rosenblatt, Distribution free tests of independence based on the sample distribution function, Ann. Math. Statist. 32 (1961), pp. 485–498.
   Google Scholar
• S. Borovkova, R. Burton, and H. Dehling, Consistency of the takens estimator for the correlation dimension, Ann. Appl. Probab. 9(2) (1999), pp. 376–390.
   Web of Science ®Google Scholar
• Y.V. Borovskikh, U-statistics in Banach spaces, VSP, Utrecht, 1996.
   Google Scholar
• S. Bouzebda, Bootstrap de l'estimateur de Hill: théorèmes limites, Ann. I.S.U.P. 54(1-2) (2010a), pp. 61–72.
   Google Scholar
• S. Bouzebda, Strong approximation of the smoothed Q-Q processes, Far East J. Theor. Stat. 31(2) (2010b), pp. 169–191.
   Google Scholar
• S. Bouzebda, On the strong approximation of bootstrapped empirical copula processes with applications, Math. Methods Stat. 21(3) (2012), pp. 153–188.
   Google Scholar
• S. Bouzebda, Asymptotic properties of pseudo maximum likelihood estimators and test in semi-parametric copula models with multiple change points, Math. Methods Stat. 23(1) (2014a), pp. 38–65.
   Google Scholar
• S. Bouzebda, General tests of independence based on empirical processes indexed by functions, Stat. Methodol. 21 (2014b), pp. 59–87.
   Google Scholar
• S. Bouzebda, I. Elhattab, and B. Nemouchi, On the uniform-in-bandwidth consistency of the general conditional U-statistics based on the copula representation, J. Nonparametr. Stat. 33(2) (2021), pp. 321–358.
   Web of Science ®Google Scholar
• S. Bouzebda, A.A. Ferfache, and T. El-hadjali, Uniform in bandwidth consistency of conditional U-statistics adaptive to intrinsic dimension in presence of censored data, Sankhya A. 85(2) (2023), pp. 1548–1606.
   Google Scholar
• S. Bouzebda, General tests of conditional independence based on empirical processes indexed by functions, Jpn. J. Stat. Data Sci. 6(1) (2023a), pp. 115–177.
   Google Scholar
• S. Bouzebda, On the weak convergence and the uniform-in-bandwidth consistency of the general conditional U-processes based on the copula representation: multivariate setting, Hacet. J. Math. Stat.52(5) (2023b), pp. 1303–1348.
   Web of Science ®Google Scholar
• S. Bouzebda and M. Chaouch, Uniform limit theorems for a class of conditional Z-estimators when covariates are functions, J. Multivariate Anal. 189 (2022), pp. 104872.
   Web of Science ®Google Scholar
• S. Bouzebda and M. Cherfi, General bootstrap for dual ϕ-divergence estimates, J. Probab. Stat. (2012), pp. 33. pages Art. ID 834107.
   Google Scholar
• S. Bouzebda and S. Didi, Multivariate wavelet density and regression estimators for stationary and ergodic discrete time processes: asymptotic results, Comm. Stat. Theory Methods 46(3) (2017), pp. 1367–1406.
   Web of Science ®Google Scholar
• S. Bouzebda and A.A. Ferfache, Functional central limit theorems for triangular arrays of function-indexed U-processes under uniformly integrable entropy conditions, Submitted (2023), pp. 1–41.
   Google Scholar
• S. Bouzebda and N. Limnios, On general bootstrap of empirical estimator of a semi-Markov kernel with applications, J. Multivariate Anal. 116 (2013), pp. 52–62.
   Web of Science ®Google Scholar
• S. Bouzebda and B. Nemouchi, Central limit theorems for conditional empirical and conditional U-processes of stationary mixing sequences, Math. Methods Stat. 28(3) (2019), pp. 169–207.
   Web of Science ®Google Scholar
• S. Bouzebda and B. Nemouchi, Weak-convergence of empirical conditional processes and conditional U-processes involving functional mixing data, Stat. Inference Stoch. Process. 26(1) (2023), pp. 33–88.
   Google Scholar
• S. Bouzebda and A. Nezzal, Uniform consistency and uniform in number of neighbors consistency for nonparametric regression estimates and conditional U-statistics involving functional data, Jpn. J. Stat. Data Sci. 5(2) (2022), pp. 431–533.
   Google Scholar
• S. Bouzebda and A. Nezzal, Asymptotic properties of conditional U-statistics using delta sequences, Comm. Statist. Theory Methods (2024a), pp. 1–56.
   Web of Science ®Google Scholar
• S. Bouzebda and A. Nezzal, Uniform in number of neighbors consistency and weak convergence of kNN empirical conditional processes and kNN conditional U-processes involving functional mixing data, AIMS Math. 9(2) (2024b), pp. 4427–4550.
   Web of Science ®Google Scholar
• S. Bouzebda and I. Soukarieh, Renewal type bootstrap for U-process Markov chains, Markov Process. Relat. Fields 28(5) (2022), pp. 673–735.
   Web of Science ®Google Scholar
• S. Bouzebda and I. Soukarieh, Non-parametric conditional U-processes for locally stationary functional random fields under stochastic sampling design, Mathematics 11(1) (2023), pp. 1–70.
   Web of Science ®Google Scholar
• S. Bouzebda and N. Taachouche, On the variable bandwidth kernel estimation of conditional U-statistics at optimal rates in sup-norm, Phys. A. 625 (2023), pp. 129000.
   Web of Science ®Google Scholar
• S. Bouzebda and N. Taachouche, Rates of the strong uniform consistency with rates for conditional U-statistics estimators with general kernels on manifolds, Math. Methods Stat. 33(1) (2024), pp. 1–55.
   Google Scholar
• J. Bretagnolle, Lois limites du bootstrap de certaines fonctionnelles, Ann. Inst. H. Poincaré Sect. B (N.S.) 19(3) (1983), pp. 281–296.
   Google Scholar
• Q. Cao, Z.-C. Guo, and Y. Ying, Generalization bounds for metric and similarity learning, Mach. Learn. 102(1) (2016), pp. 115–132.
   Web of Science ®Google Scholar
• L. Chen, A.T.K. Wan, S. Zhang, and Y. Zhou, Distributed algorithms for U-statistics-based empirical risk minimization, J. Mach. Learn. Res. 24 (2023), pp. 43. Paper No. 263.
   Web of Science ®Google Scholar
• R.F. Cintra, M. Valk, and D. Marcondes Filho, A model-free-based control chart for batch process using u-statistics, J. Process. Control. 132 (2023). Cited by: 0.
   Web of Science ®Google Scholar
• S. Clémençon, G. Lugosi, and N. Vayatis, Ranking and empirical minimization of U-statistics, Ann. Stat. 36(2) (2008), pp. 844–874.
   Web of Science ®Google Scholar
• S. Clémençon, A statistical view of clustering performance through the theory of U-processes, J. Multivariate Anal. 124 (2014), pp. 42–56.
   Web of Science ®Google Scholar
• S. Clémençon, I. Colin, and A. Bellet, Scaling-up empirical risk minimization: optimization of incomplete U-statistics, J. Mach. Learn. Res. 17 (2016), pp. 36. Paper No. 76.
   Web of Science ®Google Scholar
• M. Cuparić, Asymptotic properties of inverse probability of censored weighted U-empirical process for right-censored data with applications, Statistics 55(5) (2021), pp. 1035–1057.
   Web of Science ®Google Scholar
• M. Cuparić and B. Milošević, New characterization-based exponentiality tests for randomly censored data, TEST 31(2) (2022), pp. 461–487.
   Web of Science ®Google Scholar
• G.B. Cybis, M. Valk, and S.R.C. Lopes, Clustering and classification problems in genetics through U-statistics, J. Stat. Comput. Simul. 88(10) (2018), pp. 1882–1902.
   Web of Science ®Google Scholar
• L. Dümbgen, Limit theorems for the simplicial depth, Statist. Probab. Lett. 14(2) (1992), pp. 119–128.
   Web of Science ®Google Scholar
• V.H. de la Peña and E. Giné, Decoupling. Probability and its Applications (New York). Springer-Verlag, New York. From dependence to independence, Randomly stopped processes. U-statistics and processes. Martingales and beyond, 1999.
   Google Scholar
• R.M. Dudley, Central limit theorems for empirical measures, Ann. Probab. 6(6) (1978), pp. 899–929.
   Web of Science ®Google Scholar
• R.M. Dudley, Uniform Central Limit Theorems, Volume 142 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, New York second edition, 2014.
   Google Scholar
• R.M. Dudley, A course on empirical processes, in École d'été de probabilités de Saint-Flour, XII–1982, volume 1097 of Lecture Notes in Math., Springer, Berlin, 1984, pp. 1–142.
   Google Scholar
• E.B. Dynkin and A. Mandelbaum, Symmetric statistics, poisson point processes, and multiple wiener integrals, Ann. Stat. 11(3) (1983), pp. 739–745.
   Web of Science ®Google Scholar
• L. Faivishevsky and J. Goldberger, Ica based on a smooth estimation of the differential entropy, in D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, editors, Advances in Neural Information Processing Systems, volume 21. Curran Associates, Inc, 2009.
   Google Scholar
• A.A. Filippova, Mises theorem on the limit behaviour of functionals derived from empirical distribution functions, Dokl. Akad. Nauk SSSR 129 (1959), pp. 44–47.
   Google Scholar
• E.W. Frees, Infinite order U-statistics, Scand. J. Stat. 16(1) (1989), pp. 29–45.
   Web of Science ®Google Scholar
• S.S. Ghannadpour, S.E. Kalkhoran, H. Jalili, and M. Behifar, Delineation of mineral potential zone using u-statistic method in processing satellite remote sensing images, Int. J. Min. Geo-Eng. 57(4) (2023), pp. 445–453.
   Google Scholar
• S. Ghosal, A. Sen, and A.W. van der Vaart, Testing monotonicity of regression, Ann. Stat. 28(4) (2000), pp. 1054–1082.
   Web of Science ®Google Scholar
• E. Giné and D.M. Mason, Laws of the iterated logarithm for the local U-statistic process, J. Theoret. Probab. 20(3) (2007), pp. 457–485.
   Web of Science ®Google Scholar
• E. Giné and J. Zinn, Some limit theorems for empirical processes, Ann. Probab. 12(4) (1984), pp. 929–998. With discussion.
   Web of Science ®Google Scholar
• E. Giné and J. Zinn, Bootstrapping general empirical measures, Ann. Probab. 18(2) (1990), pp. 851–869.
   Web of Science ®Google Scholar
• E. Giné and J. Zinn, Lectures on the central limit theorem for empirical processes, in Probability and Banach spaces (Zaragoza, 1985), volume 1221 of Lecture Notes in Math., Springer, Berlin, 1986, pp. 50–113.
   Google Scholar
• P.R. Halmos, The theory of unbiased estimation, Ann. Math. Stat. 17 (1946), pp. 34–43.
   Google Scholar
• Q. Han, Multiplier U-processes: sharp bounds and applications, Bernoulli 28(1) (2022), pp. 87–124.
   Web of Science ®Google Scholar
• F. Han and T. Qian, On inference validity of weighted U-statistics under data heterogeneity, Electron. J. Stat. 12(2) (2018), pp. 2637–2708.
   Web of Science ®Google Scholar
• G.H. Hardy, Divergent series. Éditions Jacques Gabay, Sceaux. With a preface by J. E. Littlewood and a note by L. S. Bosanquet, Reprint of the revised (1963) edition, 1992.
   Google Scholar
• T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning, 2nd ed., Springer Series in Statistics, Springer, New York, 2009. Data mining, inference, and prediction.
   Google Scholar
• C. Heilig and D. Nolan, Limit theorems for the infinite-degree U-process, Stat. Sin. 11(1) (2001), pp. 289–302.
   Web of Science ®Google Scholar
• R. Helmers, P. Janssen, and R. Serfling, Glivenko-Cantelli properties of some generalized empirical DF's and strong convergence of generalized L-statistics, Probab. Theory Relat. Fields 79(1) (1988), pp. 75–93.
   Web of Science ®Google Scholar
• W. Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math. Stat. 19 (1948), pp. 293–325.
   Google Scholar
• T. Hsing and W.B. Wu, On weighted U-statistics for stationary processes, Ann. Probab. 32(2) (2004), pp. 1600–1631.
   Web of Science ®Google Scholar
• B. Huang, Y. Liu, and L. Peng, Distributed inference for two-sample U-statistics in massive data analysis, Scand. J. Stat. 50(3) (2023), pp. 1090–1115.
   Web of Science ®Google Scholar
• S.R. Jammalamadaka and S. Janson, Limit theorems for a triangular scheme of U-statistics with applications to inter-point distances, Ann. Probab. 14(4) (1986), pp. 1347–1358.
   Web of Science ®Google Scholar
• S. Janson, A functional limit theorem for random graphs with applications to subgraph count statistics, Random Struct. Algorithms 1(1) (1990), pp. 15–37.
   Google Scholar
• S. Janson, Asymptotic normality for m-dependent and constrained U-statistics, with applications to pattern matching in random strings and permutations, Adv. in Appl. Probab. 55(3) (2023), pp. 841–894.
   Google Scholar
• R. Jin, S. Wang, and Y. Zhou, Regularized distance metric learning:theory and algorithm, in Y. Bengio, D. Schuurmans, J. Lafferty, C. Williams, and A. Culotta, editors, Advances in Neural Information Processing Systems, volume 22. Curran Associates, Inc, 2009.
   Google Scholar
• E. Joly and G. Lugosi, Robust estimation of U-statistics, Stoch. Process. Appl. 126(12) (2016), pp. 3760–3773.
   Web of Science ®Google Scholar
• M. Kac, On deviations between theoretical and empirical distributions, Proc. Nat. Acad. Sci. U.S.A.35 (1949), pp. 252–257.
   PubMed Web of Science ®Google Scholar
• I. Kim and A. Ramdas, Dimension-agnostic inference using cross U-statistics, Bernoulli 30(1) (2024), pp. 683–711.
   Google Scholar
• V.S. Koroljuk and Y.V. Borovskich, Theory of U-statistics, volume 273 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht. Translated from the 1989 Russian original by P. V. Malyshev and D. V. Malyshev and revised by the authors, 1994.
   Google Scholar
• M.R. Kosorok, Bootstrapping in Grenander estimator, in Beyond parametrics in interdisciplinary research Festschrift in honor of Professor Pranab K. Sen, volume 1 of Inst. Math. Stat. (IMS) Collect., Inst. Math. Statist., Beachwood, OH, 2008, pp. 282–292.
   Google Scholar
• L. Le Cam, Remarques sur le théorème limite central dans les espaces localement convexes, in Les probabilités sur les structures algébriques (Actes Colloq. Internat. CNRS, No. 186, Clermont-Ferrand, 1969), pages 233–249. Éditions Centre Nat. Recherche Sci., Paris. Avec commentaire en anglais par R. M. Dudley, 1970.
   Google Scholar
• T. Le Minh, U-statistics on bipartite exchangeable networks, ESAIM Probab. Stat. 27 (2023), pp. 576–620.
   Web of Science ®Google Scholar
• M. Ledoux and M. Talagrand, Conditions d'intégrabilité pour les multiplicateurs dans le TLC banachique, Ann. Probab. 14(3) (1986), pp. 916–921.
   Google Scholar
• M. Ledoux and M. Talagrand, Probability in Banach Spaces, Classics in Mathematics, Springer-Verlag, Berlin, 2011. Isoperimetry and processes, Reprint of the 1991 edition.
   Google Scholar
• A.J. Lee, U-statistics, Volume 110 of Statistics: Textbooks and Monographs, Marcel Dekker Inc., New York, 1990. Theory and practice.
   Google Scholar
• H. Li, C. Ren, and L. Li, U-processes and preference learning, Neural Comput. 26(12) (2014), pp. 2896–2924.
   PubMed Web of Science ®Google Scholar
• H.W. Lilliefors, On the Kolmogorov–Smirnov Test for Normality with Mean and Variance Unknown, J. Am. Stat. Assoc. 62(318) (1967), pp. 399–402.
   Web of Science ®Google Scholar
• F. Lim and V.M. Stojanovic, On U-statistics and compressed sensing I: non-asymptotic average-case analysis, IEEE Trans. Signal Process. 61(10) (2013), pp. 2473–2485.
   Web of Science ®Google Scholar
• R.Y. Liu, On a notion of data depth based on random simplices, Ann. Stat. 18(1) (1990), pp. 405–414.
   Web of Science ®Google Scholar
• Q. Liu, J. Lee, and M. Jordan, A kernelized stein discrepancy for goodness-of-fit tests, in M. F. Balcan and K. Q. Weinberger, editors, Proceedings of The 33rd International Conference on Machine Learning, volume 48 of Proceedings of Machine Learning Research, New York, New York, USA. PMLR, 2016, pp. 276–284.
   Google Scholar
• M. Loève, Probability Theory. II, 4th ed., Graduate Texts in Mathematics, Vol. 46, Springer-Verlag, New York-Heidelberg, 1978.
   Google Scholar
• P. Major, Asymptotic distributions for weighted U-statistics, Ann. Probab. 22(3) (1994), pp. 1514–1535.
   Web of Science ®Google Scholar
• D. Nolan and D. Pollard, U-processes: rates of convergence, Ann. Stat. 15(2) (1987), pp. 780–799.
   Web of Science ®Google Scholar
• D. Nolan and D. Pollard, Functional limit theorems for U-processes, Ann. Probab. 16(3) (1988), pp. 1291–1298.
   Web of Science ®Google Scholar
• K.A. O'Neil and R.A. Redner, Asymptotic distributions of weighted U-statistics of degree 2, Ann. Probab. 21(2) (1993), pp. 1159–1169.
   Web of Science ®Google Scholar
• W. Peng, T. Coleman, and L. Mentch, Rates of convergence for random forests via generalized U-statistics, Electron. J. Stat. 16(1) (2022), pp. 232–292.
   Web of Science ®Google Scholar
• D. Pollard, Convergence of Stochastic processes, Springer Series in Statistics, Springer-Verlag, New York, 1984.
   Google Scholar
• D. Pollard, Empirical Processes: theory and Applications, Volume 2 of NSF-CBMS Regional Conference Series in Probability and Statistics, Institute of Mathematical Statistics, Hayward, CA; American Statistical Association, Alexandria, VA, 1990.
   Google Scholar
• J. Præstgaard and J.A. Wellner, Exchangeably weighted bootstraps of the general empirical process, Ann. Probab. 21(4) (1993), pp. 2053–2086.
   Web of Science ®Google Scholar
• L. Rüschendorf, Weak convergence of the weighted multiparameter empirical process, in Nonparametric asymptotic statistics (Proc. Conf., Rouen, 1979) (French), volume 821 of Lecture Notes in Math., Springer, Berlin, 1980, pp. 86–94.
   Google Scholar
• M. Rifi and F. Utzet, On the asymptotic behavior of weighted U-statistics, J. Theoret. Probab. 13(1) (2000), pp. 141–167.
   Web of Science ®Google Scholar
• H. Rubin and R.A. Vitale, Asymptotic distribution of symmetric statistics, Ann. Stat. 8(1) (1980), pp. 165–170.
   Web of Science ®Google Scholar
• A. Schick, Y. Wang, and W. Wefelmeyer, Tests for normality based on density estimators of convolutions, Stat. Probab. Lett. 81(2) (2011), pp. 337–343.
   Web of Science ®Google Scholar
• R.J. Serfling, Approximation Theorems of Mathematical Statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons Inc, New York, 1980.
   Google Scholar
• C.P. Shapiro and L. Hubert, Asymptotic normality of permutation statistics derived from weighted sums of bivariate functions, Ann. Stat. 7(4) (1979), pp. 788–794.
   Web of Science ®Google Scholar
• R.P. Sherman, Maximal inequalities for degenerate U-processes with applications to optimization estimators, Ann. Stat. 22(1) (1994), pp. 439–459.
   Web of Science ®Google Scholar
• G.R. Shorack, The weighted empirical process of row independent random variables with arbitrary distribution functions, Stat. Neerl. 33(4) (1979), pp. 169–189.
   Google Scholar
• B.W. Silverman, Distances on circles, toruses and spheres, J. Appl. Probab. 15(1) (1978), pp. 136–143.
   Web of Science ®Google Scholar
• Y. Song, X. Chen, and K. Kato, Approximating high-dimensional infinite-order U-statistics: statistical and computational guarantees, Electron. J. Stat. 13(2) (2019), pp. 4794–4848.
   Web of Science ®Google Scholar
• I. Soukarieh and S. Bouzebda, Exchangeably weighted bootstraps of general Markov U-process, Mathematics 10(20) (2022), pp. 1–42.
   Web of Science ®Google Scholar
• I. Soukarieh and S. Bouzebda, Renewal type bootstrap for increasing degree U-process of a Markov chain, J. Multivariate Anal. 195 (2023), pp. 105143.
   Web of Science ®Google Scholar
• I. Soukarieh and S. Bouzebda, Weak convergence of the conditional U-statistics for locally stationary functional time series, Stat. Inference Stoch. Process. (2024), pp. 1–78.
   Google Scholar
• W. Stute, Almost sure representations of the product-limit estimator for truncated data, Ann. Stat.21(1) (1993), pp. 146–156.
   Web of Science ®Google Scholar
• K.K. Sudheesh, S. Anjana, and M. Xie, U-statistics for left truncated and right censored data, Statistics 57(4) (2023), pp. 900–917.
   Web of Science ®Google Scholar
• A van der Vaart, Asymptotic Statistics, Volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 1998.
   Google Scholar
• A. van der Vaart and J.A. Wellner, Weak Convergence and Empirical Processes, Springer Series in Statistics, Springer-Verlag, New York, 1996.
   Google Scholar
• A.W. van der Vaart and J.A. Wellner, Weak Convergence and Empirical Processes – with Applications to Statistics, 2nd ed., Springer Series in Statistics, Springer, Cham, 2023. [of 1385671].
   Google Scholar
• D. Varron, Donsker and Glivenko-Cantelli theorems for a class of processes generalizing the empirical process, Electron. J. Stat. 8(2) (2014), pp. 2296–2320.
   Google Scholar
• R. von Mises, On the asymptotic distribution of differentiable statistical functions, Ann. Math. Stat.18 (1947), pp. 309–348.
   Google Scholar
• E. Yilmaz, J.A. Aslam, and S. Robertson, A new rank correlation coefficient for information retrieval, in Proceedings of the 31st Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, SIGIR '08, New York, NY, USA. Association for Computing Machinery, 2008, pp. 587–594.
   Google Scholar
• D. Zhang, Bayesian bootstraps for U-processes, hypothesis tests and convergence of Dirichlet U-processes, Stat. Sin. 11(2) (2001), pp. 463–478.
   Web of Science ®Google Scholar
• Z.-S. Zhang, Berry-Esseen bounds for generalized U-statistics, Electron. J. Probab. 27 (2022), pp. 36. Paper No. 134.
   Web of Science ®Google Scholar