Bernoulli

  • Bernoulli
  • Volume 20, Number 2 (2014), 846-877.

Continuous mapping approach to the asymptotics of $U$- and $V$-statistics

Eric Beutner and Henryk Zähle

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Abstract

We derive a new representation for $U$- and $V$-statistics. Using this representation, the asymptotic distribution of $U$- and $V$-statistics can be derived by a direct application of the Continuous Mapping theorem. That novel approach not only encompasses most of the results on the asymptotic distribution known in literature, but also allows for the first time a unifying treatment of non-degenerate and degenerate $U$- and $V$-statistics. Moreover, it yields a new and powerful tool to derive the asymptotic distribution of very general $U$- and $V$-statistics based on long-memory sequences. This will be exemplified by several astonishing examples. In particular, we shall present examples where weak convergence of $U$- or $V$-statistics occurs at the rate $a_{n}^{3}$ and $a_{n}^{4}$, respectively, when $a_{n}$ is the rate of weak convergence of the empirical process. We also introduce the notion of asymptotic (non-) degeneracy which often appears in the presence of long-memory sequences.

Article information

Source
Bernoulli, Volume 20, Number 2 (2014), 846-877.

Dates
First available in Project Euclid: 28 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.bj/1393594008

Digital Object Identifier
doi:10.3150/13-BEJ508

Mathematical Reviews number (MathSciNet)
MR3178520

Zentralblatt MATH identifier
1303.60019

Keywords
Appell polynomials central and non-central weak limit theorems empirical process Hoeffding decomposition non-degenerate and degenerate $U$- and $V$-statistics strong limit theorems strongly dependent data von Mises decomposition weakly dependent data

Citation

Beutner, Eric; Zähle, Henryk. Continuous mapping approach to the asymptotics of $U$- and $V$-statistics. Bernoulli 20 (2014), no. 2, 846--877. doi:10.3150/13-BEJ508. https://projecteuclid.org/euclid.bj/1393594008


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References

  • [1] Andersen, N.T., Giné, E. and Zinn, J. (1988). The central limit theorem for empirical processes under local conditions: The case of Radon infinitely divisible limits without Gaussian component. Trans. Amer. Math. Soc. 308 603–635.
  • [2] Arcones, M.A. and Giné, E. (1992). On the bootstrap of $U$ and $V$ statistics. Ann. Statist. 20 655–674.
  • [3] Avram, F. and Taqqu, M.S. (1987). Noncentral limit theorems and Appell polynomials. Ann. Probab. 15 767–775.
  • [4] Beutner, E., Wu, W.B. and Zähle, H. (2012). Asymptotics for statistical functionals of long-memory sequences. Stochastic Process. Appl. 122 910–929.
  • [5] Beutner, E. and Zähle, H. (2010). A modified functional delta method and its application to the estimation of risk functionals. J. Multivariate Anal. 101 2452–2463.
  • [6] Beutner, E. and Zähle, H. (2012). Deriving the asymptotic distribution of U- and V-statistics of dependent data using weighted empirical processes. Bernoulli 18 803–822.
  • [7] Beutner, E. and Zähle, H. (2014). Supplement to “Continuous mapping approach to the asymptotics of $U$- and $V$-statistics.” DOI:10.3150/13-BEJ508SUPP.
  • [8] Chen, X. and Fan, Y. (2006). Estimation of copula-based semiparametric time series models. J. Econometrics 130 307–335.
  • [9] Dehling, H. (2006). Limit theorems for dependent $U$-statistics. In Dependence in Probability and Statistics. Lecture Notes in Statist. 187 65–86. New York: Springer.
  • [10] 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.
  • [11] Dehling, H. and Taqqu, M.S. (1991). Bivariate symmetric statistics of long-range dependent observations. J. Statist. Plann. Inference 28 153–165.
  • [12] Denker, M. (1985). Asymptotic Distribution Theory in Nonparametric Statistics. Advanced Lectures in Mathematics. Braunschweig: Friedr. Vieweg & Sohn.
  • [13] Ghorpade, S.R. and Limaye, B.V. (2010). A Course in Multivariable Calculus and Analysis. Undergraduate Texts in Mathematics. New York: Springer.
  • [14] Gill, R.D., van der Laan, M.J. and Wellner, J.A. (1995). Inefficient estimators of the bivariate survival function for three models. Ann. Inst. Henri Poincaré Probab. Stat. 31 545–597.
  • [15] Giraitis, L. and Surgailis, D. (1999). Central limit theorem for the empirical process of a linear sequence with long memory. J. Statist. Plann. Inference 80 81–93.
  • [16] Halmos, P.R. (1946). The theory of unbiased estimation. Ann. Math. Statist. 17 34–43.
  • [17] Ho, H.C. and Hsing, T. (1996). On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Statist. 24 992–1024.
  • [18] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 293–325.
  • [19] Hosking, J.R.M. (1981). Fractional differencing. Biometrika 68 165–176.
  • [20] Hsing, T. (2000). Linear processes, long-range dependence and asymptotic expansions. Stat. Inference Stoch. Process. 3 19–29.
  • [21] Koroljuk, V.S. and Borovskich, Y.V. (1994). Theory of $U$-statistics. Mathematics and Its Applications 273. Dordrecht: Kluwer Academic.
  • [22] Lee, A.J. (1990). $U$-Statistics: Theory and Practice. Statistics: Textbooks and Monographs 110. New York: Dekker.
  • [23] Lévy-Leduc, C., Boistard, H., Moulines, E., Taqqu, M.S. and Reisen, V.A. (2011). Asymptotic properties of $U$-processes under long-range dependence. Ann. Statist. 39 1399–1426.
  • [24] Lévy-Leduc, C., Boistard, H., Moulines, E., Taqqu, M.S. and Reisen, V.A. (2011). Large sample behaviour of some well-known robust estimators under long-range dependence. Statistics 45 59–71.
  • [25] Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics 44. Cambridge: Cambridge Univ. Press.
  • [26] Pollard, D. (1984). Convergence of Stochastic Processes. Springer Series in Statistics. New York: Springer.
  • [27] Rio, E. (1995). A maximal inequality and dependent Marcinkiewicz–Zygmund strong laws. Ann. Probab. 23 918–937.
  • [28] Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics. New York: Wiley.
  • [29] Shao, Q.M. and Yu, H. (1996). Weak convergence for weighted empirical processes of dependent sequences. Ann. Probab. 24 2098–2127.
  • [30] Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
  • [31] van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge: Cambridge Univ. Press.
  • [32] Veillette, M.S. and Taqqu, M.S. (2013). Properties and numerical evaluation of the Rosenblatt distribution. Bernoulli 19 982–1005.
  • [33] von Mises, R. (1947). On the asymptotic distribution of differentiable statistical functions. Ann. Math. Statist. 18 309–348.
  • [34] Wu, W.B. (2003). Empirical processes of long-memory sequences. Bernoulli 9 809–831.
  • [35] Wu, W.B. (2006). Unit root testing for functionals of linear processes. Econometric Theory 22 1–14.
  • [36] Wu, W.B. (2008). Empirical processes of stationary sequences. Statist. Sinica 18 313–333.
  • [37] Yukich, J.E. (1992). Weak convergence of smoothed empirical processes. Scand. J. Stat. 19 271–279.
  • [38] Zähle, H. (2014). Marcinkiewicz-Zygmund and ordinary strong laws for empirical distribution functions and plug-in estimators. Statistics. To appear.

Supplemental materials

  • Supplementary material: Supplement to paper “Continuous mapping approach to the asymptotics of U- and V-statistics”. The supplement Beutner and Zähle [7] contains a discussion of some extensions and limitations of the approach presented in this paper.