Skip to main content

Central Limit Theorems for Conditional Empirical and Conditional U-Processes of Stationary Mixing Sequences

Abstract

In this paper we are concerned with the weak convergence to Gaussian processes of conditional empirical processes and conditional U-processes from stationary β-mixing sequences indexed by classes of functions satisfying some entropy conditions. We obtain uniform central limit theorems for conditional empirical processes and conditional U-processes when the classes of functions are uniformly bounded or unbounded with envelope functions satisfying some moment conditions. We apply our results to introduce statistical tests for conditional independence that are multivariate conditional versions of the Kendall statistics.

This is a preview of subscription content, access via your institution.

References

  1. J. Abrevaya and W. Jiang, “A Nonparametric Approach to Measuring and Testing Curvature”, J. Bus. Econom. Statist. 23 (1), 1–19 (2005).

    MathSciNet  Article  Google Scholar 

  2. H. Akaike, “An Approximation to the Density Function”, Ann. Inst. Statist. Math., Tokyo 6, 127–132 (1954).

    MathSciNet  MATH  Article  Google Scholar 

  3. M. A. Arcones and E. Giné, “Limit Theorems for U-Processes”, Ann. Probab. 21 (3), 1494–1542 (1993).

    MathSciNet  MATH  Article  Google Scholar 

  4. M. A. Arcones and E. Giné, “On the Law of the Iterated Logarithm for Canonical U-Statistics and Processes”, Stochastic Process. Appl. 58 (2), 217–245 (1995).

    MathSciNet  MATH  Article  Google Scholar 

  5. M. A. Arcones and Y. Wang, “Some New Tests for Normality Based on U-Processes”, Statist. Probab. Lett. 76 (1), 69–82 (2006).

    MathSciNet  MATH  Article  Google Scholar 

  6. M. A. Arcones and B. Yu, “Central Limit Theorems for Empirical and U-Processes of Stationary Mixing Sequences”, J. Theoret. Probab. 7 (1), 47–71 (1994).

    MathSciNet  MATH  Article  Google Scholar 

  7. M. A. Arcones, Z. Chen, and E. Giné, “Estimators Related to U-Processes with Applications to Multivariate Medians: Asymptotic Normality”, Ann. Statist. 22 (3), 1460–1477 (1994).

    MathSciNet  MATH  Article  Google Scholar 

  8. S. Borovkova, R. Burton, and H. Dehling, “Limit Theorems for Functionals of Mixing Processes with Applications to U-Statistics and Dimension Estimation”, Trans. Amer. Math. Soc. 353 (11), 4261–4318 (2001).

    MathSciNet  MATH  Article  Google Scholar 

  9. Y. V. Borovskikh, U-Statistics in Banach Spaces (VSP, Utrecht, 1996).

    MATH  Google Scholar 

  10. R. C. Bradley, “Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions”, Probab. Surv. 2, 107–144 (2005). Update of and a supplement to the 1986 original.

    MathSciNet  MATH  Article  Google Scholar 

  11. V. H. de la Peña, “Decoupling and Khintchin’s Inequalities for U-Statistics”, Ann. Probab. 20 (4), 1877–1892 (1992).

    MathSciNet  MATH  Article  Google Scholar 

  12. V. H. de la Peña and E. Giné, Decoupling, in Probability and its Applications (New York) (Springer-Verlag, New York, 1999). From dependence to independence. Randomly stopped processes. U-statistics and processes. Martingales and beyond.

    Google Scholar 

  13. P. Deheuvels, “One Bootstrap Suffices to Generate Sharp Uniform Bounds in Functional Estimation”, Kybernetika (Prague) 47 (6), 855–865 (2011).

    MathSciNet  MATH  Google Scholar 

  14. P. Deheuvels and D. M. Mason, “General Asymptotic Confidence Bands Based on Kernel-Type Function Estimators”, Statist. Inference Stoch. Process. 7 (3), 225–277 (2004).

    MathSciNet  MATH  Article  Google Scholar 

  15. M. Denker and G. Keller, “On U-Statistics and von Mises’ Statistics for Weakly Dependent Processes”, Z. Wahrsch. Verw. Gebiete 64 (4), 505–522 (1983).

    MathSciNet  MATH  Article  Google Scholar 

  16. J. Dony and D. M. Mason, “Uniform in Bandwidth Consistency of Conditional U-Statistics”, Bernoulli 14 (4), 1108–1133 (2008).

    MathSciNet  MATH  Article  Google Scholar 

  17. P. Doukhan, P. Massart, and E. Rio, “The Functional Central Limit Theorem for Strongly Mixing Processes”, Ann. Inst. H. Poincaré Probab. Statist. 30 (1), 63–82 (1994).

    MathSciNet  MATH  Google Scholar 

  18. R. M. Dudley, “The Sizes of Compact Subsets of Hilbert Space and Continuity of Gaussian Processes”, J. Functional Analysis 1, 290–330 (1967).

    MathSciNet  MATH  Article  Google Scholar 

  19. R. M. Dudley, Uniform Central Limit Theorems, in Cambridge Studies in Advanced Mathematics (Cambridge Univ. Press, Cambridge, 1999), Vol. 63.

    Book  Google Scholar 

  20. A. Dvoretzky, “Asymptotic Normality for Sums of Dependent Random Variables”, pp. 513–535 (1972).

  21. E. Eberlein, “Weak Convergence of Partial Sums of Absolutely Regular Sequences”, Statist. Probab. Lett. 2 (5), 291–293 (1984).

    MathSciNet  MATH  Article  Google Scholar 

  22. U. Einmahl and D. M. Mason, “An Empirical Process Approach to the Uniform Consistency of Kernel-Type Function Estimators”, J. Theoret. Probab. 13 (1), 1–37 (2000).

    MathSciNet  MATH  Article  Google Scholar 

  23. U. Einmahl and D. M. Mason, “Uniform in Bandwidth Consistency of Kernel-Type Function Estimators”, Ann. Statist. 33 (3), 1380–1403 (2005).

    MathSciNet  MATH  Article  Google Scholar 

  24. E. W. Frees, “Infinite Order U-Statistics”, Scand. J. Statist. 16 (1), 29–45 (1989).

    MathSciNet  MATH  Google Scholar 

  25. S. Ghosal, A. Sen, and A. W. van der Vaart, “Testing Monotonicity of Regression”, Ann. Statist. 28 (4), 1054–1082 (2000).

    MathSciNet  MATH  Article  Google Scholar 

  26. E. Giné and D. M. Mason, “Laws of the Iterated Logarithm for the Local U-Statistic Process”, J. Theoret. Probab. 20(3), 457–485 (2007a).

    MathSciNet  MATH  Article  Google Scholar 

  27. E. Giné and D. M. Mason, “On Local U-Statistic Processes and the Estimation of Densities of Functions of Several Sample Variables”, Ann. Statist. 35(3), 1105–1145 (2007b).

    MathSciNet  MATH  Article  Google Scholar 

  28. E. Giné and J. Zinn, “Some Limit Theorems for Empirical Processes”, Ann. Probab. 12 (4), 929–998 (1984). With discussion.

    MathSciNet  MATH  Article  Google Scholar 

  29. P. R. Halmos, “The Theory of Unbiased Estimation”, Ann. Math. Statist. 17, 34–43 (1946).

    MathSciNet  MATH  Article  Google Scholar 

  30. M. Harel and M. L. Puri, “Conditional U-Statistics for Dependent Random Variables”, J. Multivariate Anal. 57 (1), 84–100 (1996).

    MathSciNet  MATH  Article  Google Scholar 

  31. C. Heilig and D. Nolan, “Limit Theorems for the Infinite-Degree U-Process”, Statist. Sinica 11 (1), 289–302 (2001).

    MathSciNet  MATH  Google Scholar 

  32. W. Hoeffding, “A Class of Statistics with Asymptotically Normal Distribution”, Ann. Math. Statistics 19, 293–325 (1948).

    MathSciNet  MATH  Article  Google Scholar 

  33. J. Hoffmann-Jørgensen, “Convergence of Stochastic Processes on Polish Spaces”, (1984). Unpublished.

  34. M. Hollander and F. Proschan, “Testing Whether New is Better Than Used”, Ann. Math. Statist. 43, 1136–1146 (1972).

    MathSciNet  MATH  Article  Google Scholar 

  35. I. A. Ibragimov, “Some Limit Theorems for Stationary Processes”, Teor. Verojatnost. i Primenen. 7, 361–392 (1962).

    MathSciNet  Google Scholar 

  36. E. Joly and G. Lugosi, “Robust Estimation of U-Statistics”, Stochastic Process. Appl. 126 (12), 3760–3773 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  37. M. G. Kendall, “A New Measure of Rank Correlation”, Biometrika 30(1/2), 81–93 (1938).

    MATH  Article  Google Scholar 

  38. A. N. Kolmogorov and V. M. Tihomirov, “ε-Entropy and ε-Capacity of Sets in Functional Space”, Amer. Math. Soc. Transl. (2) 17, 277–364 (1961).

    MathSciNet  Google Scholar 

  39. V. S. Koroljuk and Y. V. Borovskich, Theory of U-Statistics, in Mathematics and its Applications (Kluwer Academic Publishers Group, Dordrecht, 1994), Vol. 273. Translated from the 1989 Russian original by P. V. Malyshev and D. V. Malyshev and revised by the authors.

    Google Scholar 

  40. M. R. Kosorok, Introduction to Empirical Processes and Semiparametric Inference, in Springer Series in Statistics (Springer, New York, 2008).

    MATH  Google Scholar 

  41. L. Le Cam, “A Remark on Empirical Measures”, In a Festschrift for Erich L. Lehmann, Wadsworth Statist./Probab. Ser., Wadsworth, Belmont, CA (1983), pp. 305–327.

  42. A. J. Lee, U-Statistics, in Statistics: Textbooks and Monographs (Marcel Dekker Inc., New York, 1990), Vol. 110. Theory and practice.

    Google Scholar 

  43. S. Lee, O. Linton, and Y.-J. Whang, “Testing for Stochastic Monotonicity”, Econometrica 77 (2), 585–602 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  44. A. Leucht, “Degenerate U- and V- Statistics under Weak Dependence: Asymptotic Theory and Bootstrap Consistency”, Bernoulli 18 (2), 552–585 (2012).

    MathSciNet  MATH  Article  Google Scholar 

  45. A. Leucht and M. H. Neumann, “Degenerate U -and V-Statistics under Ergodicity: Asymptotics, Bootstrap and Applications in Statistics”, Ann. Inst. Statist. Math. 65 (2), 349–386 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  46. E. A. Nadaraja, “On a Regression Estimate”, Teor. Verojatnost. i Primenen. 9, 157–159 (1964).

    MathSciNet  Google Scholar 

  47. D. Nolan and D. Pollard, “U-Processes: Rates of Convergence”, Ann. Statist. 15 (2), 780–799 (1987).

    MathSciNet  MATH  Article  Google Scholar 

  48. E. Parzen, “On Estimation of a Probability Density Function and Mode”, Ann. Math. Statist. 33, 1065–1076 (1962).

    MathSciNet  MATH  Article  Google Scholar 

  49. D. Pollard, Convergence of Stochastic Processes, in Springer Series in Statistics (Springer-Verlag, New York, 1984).

    Google Scholar 

  50. W. Polonik and Q. Yao, “Set-Indexed Conditional Empirical and Quantile Processes Based on Dependent Data”, J. Multivariate Anal. 80 (2), 234–255 (2002).

    MathSciNet  MATH  Article  Google Scholar 

  51. D. V. Poryvaĭ, “An Invariance Principle for Conditional Empirical Processes Formed by Dependent Random Variables”, Izv. Ross. Akad. Nauk Ser. Mat. 69 (4), 129–148 (2005).

    MathSciNet  Article  Google Scholar 

  52. B. L. S Prakasa Rao and A. Sen, “Limit Distributions of Conditional U-Statistics”, J. Theoret. Probab. 8 (2), 261–301 (1995).

    MathSciNet  MATH  Article  Google Scholar 

  53. G. Rempala and A. Gupta, “Weak Limits of U-Statistics of Infinite Order”, Random Oper. Stochastic Equations 7 (1), 39–52 (1999).

    MathSciNet  MATH  Article  Google Scholar 

  54. M. Rosenblatt, “A Central Limit Theorem and a Strong Mixing Condition”, Proc. Nat. Acad. Sci. U.S.A. 42, 43–47 (1956).

    MathSciNet  MATH  Article  Google Scholar 

  55. A. Schick, Y. Wang, and W. Wefelmeyer, “Tests for Normality Based on Density Estimators of Convolutions”, Statist. Probab. Lett. 81 (2), 337–343 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  56. A. Sen, “Uniform Strong Consistency Rates for Conditional U-Statistics”, Sankhyā Ser. A 56(2), 179–194 (1994).

    MathSciNet  MATH  Google Scholar 

  57. R. J. Serfling, Approximation Theorems of Mathematical Statistics (John Wiley & Sons, Inc., New York, 1980). Wiley Series in Probability and Mathematical Statistics.

    MATH  Book  Google Scholar 

  58. R. P. Sherman, “The Limiting Distribution of the Maximum Rank Correlation Estimator”, Econometrica 61 (1), 123–137 (1993).

    MathSciNet  MATH  Article  Google Scholar 

  59. R. P. Sherman, “Maximal Inequalities for Degenerate U-Processes with Applications to Optimization Estimators”, Ann. Statist. 22 (1), 439–459 (1994).

    MathSciNet  MATH  Article  Google Scholar 

  60. B. W. Silverman, “Distances on Circles, Toruses and Spheres”, J. Appl. Probability 15 (1), 136–143 (1978).

    MathSciNet  MATH  Article  Google Scholar 

  61. Y. Song, X. Chen, and K. Kato, Approximating High-Dimensional Infinite-Order U-Statistics: Statistical and Computational Guarantees (2019), arXiv e-prints, page arXiv:1901.01163.

  62. W. Stute, “Conditional Empirical Processes”, Ann. Statist. 14 (2), 638–647 (1986).

    MathSciNet  MATH  Article  Google Scholar 

  63. W. Stute, “Conditional U-Statistics”, Ann. Probab. 19 (2), 812–825 (1991).

    MathSciNet  MATH  Article  Google Scholar 

  64. W. Stute, “Almost Sure Representations of the Product-Limit Estimator for Truncated Data”, Ann. Statist. 21 (1), 146–156 (1993).

    MathSciNet  MATH  Article  Google Scholar 

  65. W. Stute, “Symmetrized NN-Conditional U-Statistics”, in Research Developments in Probability and Statistics, (VSP, Utrecht, 1996), pp. 231–237.

    MATH  Google Scholar 

  66. A. van der Vaart, “New Donsker Classes”, Ann. Probab. 24 (4), 2128–2140 (1996).

    MathSciNet  MATH  Article  Google Scholar 

  67. A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, in Springer Series in Statistics (Springer-Verlag, New York, 1996).

    MATH  Google Scholar 

  68. R. von Mises, On the Asymptotic Distribution of Differentiable Statistical Functions. Ann. Math. Statist. 18, 309–348 (1947).

    MathSciNet  MATH  Article  Google Scholar 

  69. G. S. Watson, “Smooth Regression Analysis”, Sankhyā Ser. A 26, 359–372 (1964).

    MathSciNet  MATH  Google Scholar 

  70. K.-i. Yoshihara, “Limiting Behavior of U-Statistics for Stationary, Absolutely Regular Processes”, Z. Wahrsch. und Verw. Gebiete 35 (3), 237–252 (1976).

    MathSciNet  MATH  Article  Google Scholar 

  71. B. Yu, “Rates of Convergence for Empirical Processes of Stationary Mixing Sequences”, Ann. Probab. 22 (1), 94–116 (1994).

    MathSciNet  MATH  Article  Google Scholar 

  72. D. Zhang, “Bayesian Bootstraps for U-Processes, Hypothesis Tests and Convergence of Dirichlet U-Processes”, Statist. Sinica 11 (2), 463–478 (2001).

    MathSciNet  MATH  Google Scholar 

  73. K. Ziegler, “On the Asymptotic Normality of Kernel Regression Estimators of the Mode in the Nonparametric Random Design Model”, J. Statist. Plann. Inference 115 (1), 123–144 (2003).

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgments

We would like to acknowledge the reviewer for detailed and useful comments which led to a more sharply focused presentation.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to S. Bouzebda or B. Nemouchi.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bouzebda, S., Nemouchi, B. Central Limit Theorems for Conditional Empirical and Conditional U-Processes of Stationary Mixing Sequences. Math. Meth. Stat. 28, 169–207 (2019). https://doi.org/10.3103/S1066530719030013

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530719030013

Keywords

  • conditional empirical processes
  • conditional U-processes
  • uniform central limit theorems
  • VC-classes
  • stationary sequence
  • absolutely regular sequences

AMS 2010 Subject Classification

  • 60F05
  • 60G15
  • 60G10
  • 62G08
  • 62G07