Electronic Journal of Statistics

A general approach to the joint asymptotic analysis of statistics from sub-samples

Stanislav Volgushev and Xiaofeng Shao

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In time series analysis, statistics based on collections of estimators computed from subsamples play a crucial role in an increasing variety of important applications. Proving results about the joint asymptotic distribution of such statistics is challenging, since it typically involves a nontrivial verification of technical conditions and tedious case-by-case asymptotic analysis. In this paper, we provide a novel technique that allows to circumvent those problems in a general setting. Our approach consists of two major steps: a probabilistic part which is mainly concerned with weak convergence of sequential empirical processes, and an analytic part providing general ways to extend this weak convergence to functionals of the sequential empirical process. Our theory provides a unified treatment of asymptotic distributions for a large class of statistics, including recently proposed self-normalized statistics and sub-sampling based p-values. In addition, we comment on the consistency of bootstrap procedures and obtain general results on compact differentiability of certain mappings that are of independent interest.

Article information

Electron. J. Statist., Volume 8, Number 1 (2014), 390-431.

First available in Project Euclid: 18 April 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62G09: Resampling methods 62G15: Tolerance and confidence regions
Secondary: 62G20: Asymptotic properties

Change point compact differentiability empirical processes self-normalization sub-sampling time series weak convergence


Volgushev, Stanislav; Shao, Xiaofeng. A general approach to the joint asymptotic analysis of statistics from sub-samples. Electron. J. Statist. 8 (2014), no. 1, 390--431. doi:10.1214/14-EJS888. https://projecteuclid.org/euclid.ejs/1397826706

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  • Adams, T. and Nobel, A. (2010). Uniform convergence of vapnik–chervonenkis classes under ergodic sampling. The Annals of Probability, 38(4):1345–1367.
  • Andrews, D. W. and Pollard, D. (1994). An introduction to functional central limit theorems for dependent stochastic processes. International Statistical Review, 62(1):119–132.
  • Aue, A. Horváth, L. and Reimherr, M. (2009). Delay times of sequential procedures for multiple time series regression models. Journal of Econometrics, 149:174–190.
  • Berkes, I., Hörmann, S., and Schauer, J. (2009). Asymptotic results for the empirical process of stationary sequences. Stochastic Processes and Their Applications, 119(4):1298–1324.
  • Beutner, E., Wu, W. B., and Zähle, H. (2012). Asymptotics for statistical functionals of long-memory sequences. Stochastic Processes and their Applications, 122(3):910–929.
  • Beutner, E. and Zähle, H. (2010). A modified functional delta method and its application to the estimation of risk functionals. Journal of Multivariate Analysis, 101(10):2452–2463.
  • Beutner, E. and Zähle, H. (2012). Deriving the asymptotic distribution of u-and v-statistics of dependent data using weighted empirical processes. Bernoulli, 18(3):803–822.
  • Billingsley, P. (1968). Convergence of Probability Measures. John Wiley & Sons Inc., New York.
  • Bücher, A. (2013). A note on weak convergence of the sequential multivariate empirical process under strong mixing. arXiv preprint arXiv:1304.5113.
  • Bücher, A., Dette, H., and Volgushev, S. (2011). New estimators of the pickands dependence function and a test for extreme-value dependence. Annals of Statistics, 39(4):1963–2006.
  • Bücher, A. and Kojadinovic, I. (2013). A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing. arXiv preprint arXiv:1306.3930.
  • Bücher, A. and Ruppert, M. (2013). Consistent testing for a constant copula under strong mixing based on the tapered block multiplier technique. Journal of Multivariate Analysis, 116(0):208 – 229.
  • Bücher, A. and Volgushev, S. (2013). Empirical and sequential empirical copula processes under serial dependence. Journal of Multivariate Analysis.
  • Bühlmann, P. L. (1993). The blockwise bootstrap in time series and empirical processes. PhD thesis, Swiss Federal Institute of Technology Zürich.
  • Chu, C-S. Stinchcombe, M. and White, H. (1995). Monitoring structural change. Econometrica, 64(5):1045–1065.
  • Csörgö, M. and Horváth, L. (1997). Limiting Theorems in Change-point Analysis. New York, Wiley.
  • Dehling, H., Mikosch, T., and Sørensen, M. (2002). Empirical process techniques for dependent data. Birkhäuser.
  • Dehling, H. and Taqqu, M. S. (1989). The empirical process of some long-range dependent sequences with an application to u-statistics. The Annals of Statistics, pages 1767–1783.
  • Doss, H. and Gill, R. D. (1992). An elementary approach to weak convergence for quantile processes, with applications to censored survival data. Journal of the American Statistical Association, 87(419):869–877.
  • Fermanian, J. D., Radulović, D., and Wegkamp, M. H. (2004). Weak convergence of empirical copula processes. Bernoulli, 10:847–860.
  • Gao, F. and Zhao, X. (2011). Delta method in large deviations and moderate deviations for estimators. The Annals of Statistics, 39(2):1211–1240.
  • Gill, R. D. and Johansen, S. (1990). A survey of product-integration with a view toward application in survival analysis. The Annals of Statistics, pages 1501–1555.
  • Hagemann, A. (2012). Stochastic equicontinuity in nonlinear time series models. Arxiv preprint arXiv:1206.2385.
  • Inoue, A. (2001). Testing for distributional change in time series. Econometric Theory, 17(1):156–187.
  • Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American statistical association, 53(282):457–481.
  • Kiefer, N. M. and Vogelsang, T. J. (2005). A new asymptotic theory for heteroskedasticity-autocorrelation robust tests. Econometric Theory, 21:1130–1164.
  • Kitamura, Y. (1997). A new asymptotic theory for heteroskedasticity-autocorrelation robust tests. Annals of Statistics, 25:2084–2102.
  • Kojadinovic, I. and Rohmer, T. (2012). Asymptotics and multiplier bootstrap of the sequential empirical copula process with applications to change-point detection. arXiv preprint arXiv:1206.2557.
  • Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer Series in Statistics, New York.
  • Künsch, H. (1989). The jackknife and the bootstrap for general stationary observations. Annals of Statistics, 17:1217–1241.
  • Lobato, I. N. (2001). Testing that a dependent process is uncorrelated. Journal of the American Statistical Association, 96:1066–1076.
  • Patton, A. J. (2009). Copula–based models for financial time series. In Handbook of financial time series, pages 767–785. Springer.
  • Perron, P. (2006). Dealing with structural breaks. Palgrave Handbook of Econometrics, vol I: Econometric Theory, eds. K. Patterson and T.C. Mills, pages 278–352.
  • Politis, D. N. and Romano, J. P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. Annals of Statistics, 22:2031–2050.
  • Politis, D. N., Romano, J. P., and Wolf, M. (1999). Subsampling. New York, Springer.
  • Radulović, D. (2009). Another look at the disjoint blocks bootstrap. Test, 18(1):195–212.
  • Rüschendorf, L. (1974). On the empirical process of multivariate, dependent random variables. Journal of Multivariate Analysis, 4(4):469–478.
  • Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. Annals of Statistics, 4:912–923.
  • Salvadori, G. (2007). Extremes in Nature: An Approach to Using Copulas, volume 56. Springer.
  • Sen, P. K. (1974). Weak convergence of multidimensional empirical processes for stationary $\phi $-mixing processes. The Annals of Probability, 2(1):147–154.
  • Shao, X. (2010a). A self-normalized approach to confidence interval construction in time series. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(3):343–366.
  • Shao, X. (2010b). corrigendum: A self-normalized approach to confidence interval construction in time series. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(5):695–696.
  • Shao, X. (2011). A simple test of changes in mean in the possible presence of long-range dependence. Journal of Time Series Analysis, 32(6):598–606.
  • Shao, X. (2012). Parametric inference in stationary time series models with dependent errors. Scandinavian Journal of Statistics, 39(4):772–783.
  • Shao, X. and Politis, D. (2013). Fixed b subsampling and the block bootstrap: improved confidence sets based on p-value calibration. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75:161–184.
  • Shao, X. and Zhang, X. (2010). Testing for change points in time series. Journal of the American Statistical Association, 105(491):1228–1240.
  • Sklar, M. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris, 8:229–231.
  • Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer Verlag, New York.
  • Vogelsang, T. J. (1999). Sources of nonmonotonic power when testing for a shift in mean of a dynamic time series. Journal of Econometrics, 88:283–299.
  • Wu, W. (2007). Strong invariance principles for dependent random variables. The Annals of Probability, 35(6):2294–2320.
  • Wu, W. and Shao, X. (2004). Limit theorems for iterated random functions. Journal of Applied Probability, 41(2):425–436.
  • Yoshihara, K.-i. (1975). Weak convergence of multidimensional empirical processes for strong mixing sequences of stochastic vectors. Probability Theory and Related Fields, 33(2):133–137.
  • Zhou, Z. and Shao, X. (2013). Inference for linear models with dependent errors. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75:323–343.