The Annals of Statistics

Asymptotic properties of U-processes under long-range dependence

C. Lévy-Leduc, H. Boistard, E. Moulines, M. S. Taqqu, and V. A. Reisen

Full-text: Open access

Abstract

Let (Xi)i≥1 be a stationary mean-zero Gaussian process with covariances $\rho(k)=\mathbb {E}(X_{1}X_{k+1})$ satisfying ρ(0) = 1 and ρ(k) = kDL(k), where D is in (0, 1), and L is slowly varying at infinity. Consider the U-process {Un(r), rI} defined as

Un(r) = 1/n(n−1) ∑1≤ijn1{G(Xi, Xj)≤r},

where I is an interval included in ℝ, and G is a symmetric function. In this paper, we provide central and noncentral limit theorems for Un. They are used to derive, in the long-range dependence setting, new properties of many well-known estimators such as the Hodges–Lehmann estimator, which is a well-known robust location estimator, the Wilcoxon-signed rank statistic, the sample correlation integral and an associated robust scale estimator. These robust estimators are shown to have the same asymptotic distribution as the classical location and scale estimators. The limiting distributions are expressed through multiple Wiener–Itô integrals.

Article information

Source
Ann. Statist., Volume 39, Number 3 (2011), 1399-1426.

Dates
First available in Project Euclid: 13 May 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1305292041

Digital Object Identifier
doi:10.1214/10-AOS867

Mathematical Reviews number (MathSciNet)
MR2850207

Zentralblatt MATH identifier
1242.62100

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62G30: Order statistics; empirical distribution functions 62G20: Asymptotic properties

Keywords
Long-range dependence U-process Hodges–Lehmann estimator Wilcoxon-signed rank test sample correlation integral

Citation

Lévy-Leduc, C.; Boistard, H.; Moulines, E.; Taqqu, M. S.; Reisen, V. A. Asymptotic properties of U -processes under long-range dependence. Ann. Statist. 39 (2011), no. 3, 1399--1426. doi:10.1214/10-AOS867. https://projecteuclid.org/euclid.aos/1305292041


Export citation

References

  • Arcones, M. A. (1994). Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 2242–2274.
  • Beran, J. (1991). M estimators of location for Gaussian and related processes with slowly decaying serial correlations. J. Amer. Statist. Assoc. 86 704–708.
  • Bickel, P. J. and Lehmann, E. L. (1979). Descriptive statistics for nonparametric models IV: Spread. In Contributions to Statistics, Hájek Memorial Volume (J. Jurečková, ed.) 33–40. Academia, Prague.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Borovkova, S., Burton, R. and Dehling, H. (2001). Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation. Trans. Amer. Math. Soc. 353 4261–4318.
  • Dehling, H. and Taqqu, M. S. (1989). The empirical process of some long-range dependent sequences with an application to U-statistics. Ann. Statist. 17 1767–1783.
  • Dehling, H. and Taqqu, M. S. (1991). Bivariate symmetric statistics of long-range dependent observations. J. Statist. Plann. Inference 28 153–165.
  • Dewan, I. and Prakasa Rao, B. L. S. (2005). Wilcoxon-signed rank test for associated sequences. Statist. Probab. Lett. 71 131–142.
  • Fox, R. and Taqqu, M. S. (1987). Multiple stochastic integrals with dependent integrators. J. Multivariate Anal. 21 105–127.
  • Grassberger, P. and Procaccia, I. (1983). Characterization of strange attractors. Phys. Rev. Lett. 50 346–349.
  • Hodges, J. L. J. and Lehmann, E. L. (1963). Estimates of location based on rank tests. Ann. Math. Statist. 34 598–611.
  • Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 293–325.
  • Hsing, T. and Wu, W. B. (2004). On weighted U-statistics for stationary processes. Ann. Probab. 32 1600–1631.
  • Lévy-Leduc, C., Boistard, H., Moulines, E., Taqqu, M. S. and Reisen, V. A. (2011a). Large sample behavior of some well-known robust estimators under long-range dependence. Statistics 45 59–71.
  • Lévy-Leduc, C., Boistard, H., Moulines, E., Taqqu, M. S. and Reisen, V. A. (2011b). Robust estimation of the scale and of the autocovariance function of Gaussian short and long-range dependent processes. J. Time Ser. Anal. 32 135–156.
  • Lévy-Leduc, C., Boistard, H., Moulines, E., Taqqu, M. S. and Reisen, V. A. (2011c). Supplement to “Asymptotic properties of U-processes under long-range dependence.” DOI:10.1214/10-AOS867SUPP.
  • Ma, Y. and Genton, M. (2000). Highly robust estimation of the autocovariance function. J. Time Ser. Anal. 21 663–684.
  • Rousseeuw, P. J. and Croux, C. (1993). Alternatives to the median absolute deviation. J. Amer. Statist. Assoc. 88 1273–1283.
  • Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
  • Shamos, M. I. (1976). Geometry and statistics: Problems at the interface. In New Directions and Recent Results in Algorithms and Complexity (J. F. Traub, ed.) 251–280. Academic Press, New York.
  • Soulier, P. (2001). Moment bounds and central limit theorem for functions of Gaussian vectors. Statist. Probab. Lett. 54 193–203.
  • Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
  • Taqqu, M. S. (1977). Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence. Z. Wahrsch. Verw. Gebiete 40 203–238.
  • van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge.
  • Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin 1 80–83.

Supplemental materials

  • Supplementary material: Proofs of Lemmas 9, 10, 12, 13, 14, 15, 16, 17 and 18 and some numerical experiments. This supplement contains proofs of Lemmas 9, 10, 12, 13, 14, 15, 16, 17 and 18 and a section containing numerical experiments illustrating some results of Section 4.