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Asymptotic distribution of two-sample empirical U-quantiles with applications to robust tests for shifts in location

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  • Dehling, Herold
  • Fried, Roland

Abstract

We derive the asymptotical distributions of two-sample U-statistics and two-sample empirical U-quantiles in the case of weakly dependent data. Our results apply to observations that can be represented as functionals of absolutely regular processes, including e.g. many classical time series models as well as data from chaotic dynamical systems. Based on these theoretical results we propose a new robust nonparametric test for the two-sample location problem, which is constructed from the median of pairwise differences between the two samples. We inspect the properties of the test in the case of weakly dependent data and compare the performance with classical tests such as the t-test and Wilcoxon’s two-sample rank test with corrections for dependencies. Simulations indicate that the new test offers better power than the Wilcoxon test in case of skewed and heavy tailed distributions, when at least one of the two samples is not very large. The test is then applied for detecting shifts of location in some weakly dependent time series, which are contaminated by outliers.

Suggested Citation

  • Dehling, Herold & Fried, Roland, 2012. "Asymptotic distribution of two-sample empirical U-quantiles with applications to robust tests for shifts in location," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 124-140.
    • Handle: RePEc:eee:jmvana:v:105:y:2012:i:1:p:124-140
    DOI: 10.1016/j.jmva.2011.08.014
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    References listed on IDEAS

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    1. Dehling, Herold & Wendler, Martin, 2010. "Central limit theorem and the bootstrap for U-statistics of strongly mixing data," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 126-137, January.
    2. Fried, Roland & Einbeck, Jochen & Gather, Ursula, 2007. "Weighted Repeated Median Smoothing and Filtering," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1300-1308, December.
    3. Isha Dewan & B. Rao, 2003. "Mann-Whitney test for associated sequences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(1), pages 111-119, March.
    4. Wendler, Martin, 2011. "Bahadur representation for U-quantiles of dependent data," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1064-1079, July.
    5. Peligrad, M. & Shao, Q. M., 1995. "Estimation of the Variance of Partial Sums for [rho]-Mixing Random Variables," Journal of Multivariate Analysis, Elsevier, vol. 52(1), pages 140-157, January.
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    Cited by:

    1. Marie Hušková & Zuzana Prášková, 2014. "Comments on: Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 265-269, June.
    2. Dehling, Herold & Fried, Roland & Sharipov, Olimjon Sh. & Vogel, Daniel & Wornowizki, Max, 2013. "Estimation of the variance of partial sums of dependent processes," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 141-147.
    3. Dag Tjøstheim, 2012. "Rejoinder on: Some recent theory for autoregressive count time series," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(3), pages 469-476, September.
    4. Lajos Horváth & Gregory Rice, 2014. "Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 219-255, June.
    5. Herold Dehling, 2014. "Comments on: Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 261-264, June.

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