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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Abrevaya and W. Jiang, “A Nonparametric Approach to Measuring and Testing Curvature”, J. Bus. Econom. Statist. 23 (1), 1–19 (2005).
H. Akaike, “An Approximation to the Density Function”, Ann. Inst. Statist. Math., Tokyo 6, 127–132 (1954).
M. A. Arcones and E. Giné, “Limit Theorems for U-Processes”, Ann. Probab. 21 (3), 1494–1542 (1993).
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).
M. A. Arcones and Y. Wang, “Some New Tests for Normality Based on U-Processes”, Statist. Probab. Lett. 76 (1), 69–82 (2006).
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).
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).
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).
Y. V. Borovskikh, U-Statistics in Banach Spaces (VSP, Utrecht, 1996).
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.
V. H. de la Peña, “Decoupling and Khintchin’s Inequalities for U-Statistics”, Ann. Probab. 20 (4), 1877–1892 (1992).
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.
P. Deheuvels, “One Bootstrap Suffices to Generate Sharp Uniform Bounds in Functional Estimation”, Kybernetika (Prague) 47 (6), 855–865 (2011).
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).
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).
J. Dony and D. M. Mason, “Uniform in Bandwidth Consistency of Conditional U-Statistics”, Bernoulli 14 (4), 1108–1133 (2008).
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).
R. M. Dudley, “The Sizes of Compact Subsets of Hilbert Space and Continuity of Gaussian Processes”, J. Functional Analysis 1, 290–330 (1967).
R. M. Dudley, Uniform Central Limit Theorems, in Cambridge Studies in Advanced Mathematics (Cambridge Univ. Press, Cambridge, 1999), Vol. 63.
A. Dvoretzky, “Asymptotic Normality for Sums of Dependent Random Variables”, pp. 513–535 (1972).
E. Eberlein, “Weak Convergence of Partial Sums of Absolutely Regular Sequences”, Statist. Probab. Lett. 2 (5), 291–293 (1984).
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).
U. Einmahl and D. M. Mason, “Uniform in Bandwidth Consistency of Kernel-Type Function Estimators”, Ann. Statist. 33 (3), 1380–1403 (2005).
E. W. Frees, “Infinite Order U-Statistics”, Scand. J. Statist. 16 (1), 29–45 (1989).
S. Ghosal, A. Sen, and A. W. van der Vaart, “Testing Monotonicity of Regression”, Ann. Statist. 28 (4), 1054–1082 (2000).
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).
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).
E. Giné and J. Zinn, “Some Limit Theorems for Empirical Processes”, Ann. Probab. 12 (4), 929–998 (1984). With discussion.
P. R. Halmos, “The Theory of Unbiased Estimation”, Ann. Math. Statist. 17, 34–43 (1946).
M. Harel and M. L. Puri, “Conditional U-Statistics for Dependent Random Variables”, J. Multivariate Anal. 57 (1), 84–100 (1996).
C. Heilig and D. Nolan, “Limit Theorems for the Infinite-Degree U-Process”, Statist. Sinica 11 (1), 289–302 (2001).
W. Hoeffding, “A Class of Statistics with Asymptotically Normal Distribution”, Ann. Math. Statistics 19, 293–325 (1948).
J. Hoffmann-Jørgensen, “Convergence of Stochastic Processes on Polish Spaces”, (1984). Unpublished.
M. Hollander and F. Proschan, “Testing Whether New is Better Than Used”, Ann. Math. Statist. 43, 1136–1146 (1972).
I. A. Ibragimov, “Some Limit Theorems for Stationary Processes”, Teor. Verojatnost. i Primenen. 7, 361–392 (1962).
E. Joly and G. Lugosi, “Robust Estimation of U-Statistics”, Stochastic Process. Appl. 126 (12), 3760–3773 (2016).
M. G. Kendall, “A New Measure of Rank Correlation”, Biometrika 30(1/2), 81–93 (1938).
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).
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.
M. R. Kosorok, Introduction to Empirical Processes and Semiparametric Inference, in Springer Series in Statistics (Springer, New York, 2008).
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.
A. J. Lee, U-Statistics, in Statistics: Textbooks and Monographs (Marcel Dekker Inc., New York, 1990), Vol. 110. Theory and practice.
S. Lee, O. Linton, and Y.-J. Whang, “Testing for Stochastic Monotonicity”, Econometrica 77 (2), 585–602 (2009).
A. Leucht, “Degenerate U- and V- Statistics under Weak Dependence: Asymptotic Theory and Bootstrap Consistency”, Bernoulli 18 (2), 552–585 (2012).
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).
E. A. Nadaraja, “On a Regression Estimate”, Teor. Verojatnost. i Primenen. 9, 157–159 (1964).
D. Nolan and D. Pollard, “U-Processes: Rates of Convergence”, Ann. Statist. 15 (2), 780–799 (1987).
E. Parzen, “On Estimation of a Probability Density Function and Mode”, Ann. Math. Statist. 33, 1065–1076 (1962).
D. Pollard, Convergence of Stochastic Processes, in Springer Series in Statistics (Springer-Verlag, New York, 1984).
W. Polonik and Q. Yao, “Set-Indexed Conditional Empirical and Quantile Processes Based on Dependent Data”, J. Multivariate Anal. 80 (2), 234–255 (2002).
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).
B. L. S Prakasa Rao and A. Sen, “Limit Distributions of Conditional U-Statistics”, J. Theoret. Probab. 8 (2), 261–301 (1995).
G. Rempala and A. Gupta, “Weak Limits of U-Statistics of Infinite Order”, Random Oper. Stochastic Equations 7 (1), 39–52 (1999).
M. Rosenblatt, “A Central Limit Theorem and a Strong Mixing Condition”, Proc. Nat. Acad. Sci. U.S.A. 42, 43–47 (1956).
A. Schick, Y. Wang, and W. Wefelmeyer, “Tests for Normality Based on Density Estimators of Convolutions”, Statist. Probab. Lett. 81 (2), 337–343 (2011).
A. Sen, “Uniform Strong Consistency Rates for Conditional U-Statistics”, Sankhyā Ser. A 56(2), 179–194 (1994).
R. J. Serfling, Approximation Theorems of Mathematical Statistics (John Wiley & Sons, Inc., New York, 1980). Wiley Series in Probability and Mathematical Statistics.
R. P. Sherman, “The Limiting Distribution of the Maximum Rank Correlation Estimator”, Econometrica 61 (1), 123–137 (1993).
R. P. Sherman, “Maximal Inequalities for Degenerate U-Processes with Applications to Optimization Estimators”, Ann. Statist. 22 (1), 439–459 (1994).
B. W. Silverman, “Distances on Circles, Toruses and Spheres”, J. Appl. Probability 15 (1), 136–143 (1978).
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.
W. Stute, “Conditional Empirical Processes”, Ann. Statist. 14 (2), 638–647 (1986).
W. Stute, “Conditional U-Statistics”, Ann. Probab. 19 (2), 812–825 (1991).
W. Stute, “Almost Sure Representations of the Product-Limit Estimator for Truncated Data”, Ann. Statist. 21 (1), 146–156 (1993).
W. Stute, “Symmetrized NN-Conditional U-Statistics”, in Research Developments in Probability and Statistics, (VSP, Utrecht, 1996), pp. 231–237.
A. van der Vaart, “New Donsker Classes”, Ann. Probab. 24 (4), 2128–2140 (1996).
A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, in Springer Series in Statistics (Springer-Verlag, New York, 1996).
R. von Mises, On the Asymptotic Distribution of Differentiable Statistical Functions. Ann. Math. Statist. 18, 309–348 (1947).
G. S. Watson, “Smooth Regression Analysis”, Sankhyā Ser. A 26, 359–372 (1964).
K.-i. Yoshihara, “Limiting Behavior of U-Statistics for Stationary, Absolutely Regular Processes”, Z. Wahrsch. und Verw. Gebiete 35 (3), 237–252 (1976).
B. Yu, “Rates of Convergence for Empirical Processes of Stationary Mixing Sequences”, Ann. Probab. 22 (1), 94–116 (1994).
D. Zhang, “Bayesian Bootstraps for U-Processes, Hypothesis Tests and Convergence of Dirichlet U-Processes”, Statist. Sinica 11 (2), 463–478 (2001).
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).
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
Corresponding authors
About this article
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
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