Journal of Advanced Analysis and Applications
Volume 1, Issue 2 (2026), Pages 1–57

Strong Approximations for Multivariate Integrated Empirical Processes:
Two-Sample Testing, Change-Point Detection, and Inference with Estimated Parameters
Author: S. Bouzebda

Université de Technologie de Compiègne, LMAC (Laboratory of Applied Mathematics of Compiègne),
Compiègne, France
Email: salim.bouzebda@utc.fr

Abstract. We investigate a broad class of multivariate \(p\)-fold integrated empirical processes obtained by iterated lower-orthant Stieltjes integration of the empirical distribution function on \([0,1]^d\). After deriving exact structural representations for the corresponding integrated distribution and empirical distribution functionals, we show that, in the scalarizable regime (including the univariate and effectively totally ordered/comonotone settings), the \(p\)-fold integrated empirical process admits a precise reduction to the ordinary multivariate empirical process, with leading term \((p!)^{-1} F(\mathbf{u})^p \alpha_n(\mathbf{u})\). In the genuinely multivariate partial-order setting, the first-order structure is governed instead by the chain functional and its influence function. This algebraic decomposition permits a systematic transfer of strong approximation results from the classical empirical process to its integrated counterparts. More specifically, on suitable probability spaces, we construct almost sure couplings of the integrated empirical process with appropriately weighted \(F\)-Brownian bridges and \(F\)-Kiefer processes, and we obtain explicit uniform approximation rates over \([0,1]^d\). These couplings yield asymptotic approximations for integrated Kolmogorov–Smirnov and Cram’er–von Mises type functionals, as well as a functional law of the iterated logarithm. We then exploit the resulting Gaussian approximations in two inferential directions: first, for multivariate two-sample procedures based on modified \(p\)-fold integrated empirical contrasts; second, for change-point statistics constructed from sequential integrated empirical discrepancies, for which asymptotically valid Gaussian calibrations are derived. We further analyze plug-in versions of the integrated empirical process in parametric settings and establish asymptotic Gaussian approximations in the presence of estimated nuisance parameters. A simulation study in dimension \(d=2\) complements the theory and illustrates the finite-sample behavior of the proposed procedures under dependence, marginal, and joint perturbation regimes.

Keywords: multivariate integrated empirical process; strong approximation; Brownian bridge; Kiefer process; law of the iterated logarithm; two-sample testing; change-point analysis; estimated parameters

MSC (2020): 60F17, 62G30, 62G20, 60F15, 62F03, 62F12

References

• R. J. Adler, An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, Institute of Mathematical Statistics Lecture Notes–Monograph Series, Vol. 12, Institute of Mathematical Statistics, Hayward, CA, 1990, 160 pp.
• S. Alvarez-Andrade, S. Bouzebda, Some nonparametric tests for change-point detection based on the P–P and Q–Q plot processes, Sequential Anal. 33 (2014), 360–399.
  DOI: https://doi.org/10.1080/07474946.2014.916930
• S. Alvarez-Andrade, S. Bouzebda, A. Lachal, Some asymptotic results for the integrated empirical process with applications to statistical tests, Comm. Statist. Theory Methods 46 (2017), 3365–3392.
  DOI: https://doi.org/10.1080/03610926.2015.1060346
• S. Alvarez-Andrade, S. Bouzebda, A. Lachal, Strong approximations for the p-fold integrated empirical process with applications to statistical tests, TEST 27 (2018), 826–849.
  DOI: https://doi.org/10.1007/s11749-017-0572-0
• M. A. Arcones, E. Giné, Limit theorems for U-processes, Ann. Probab. 21 (1993), 1494–1542.
  DOI: https://doi.org/10.1214/aop/1176989128
• A. Aue, L. Horváth, Structural breaks in time series, J. Time Series Anal. 34 (2013), 1–16.
  DOI: https://doi.org/10.1111/j.1467-9892.2012.00819.x
• P. J. Bickel, M. J. Wichura, Convergence criteria for multiparameter stochastic processes and some applications, Ann. Math. Statist. 42 (1971), 1656–1670.
  DOI: https://doi.org/10.1214/aoms/1177693164
• P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York–London–Sydney, 1968, 253 pp.
• I. S. Borisov, Approximation of empirical fields that are constructed from vector observations with dependent coordinates, Siberian Math. J. 23 (1982), 615–623.
  DOI: https://doi.org/10.1007/BF00971279
• S. Bouzebda, On the strong approximation of bootstrapped empirical copula processes with applications, Math. Methods Statist. 21 (2012), 153–188.
  DOI: https://doi.org/10.3103/S1066530712030015
• S. Bouzebda, Asymptotic properties of pseudo maximum likelihood estimators and test in semi-parametric copula models with multiple change points, Math. Methods Statist. 23 (2014), 38–65.
  DOI: https://doi.org/10.3103/S1066530714010037
• S. Bouzebda, Kac’s representation for empirical copula process from an asymptotic viewpoint, Statist. Probab. Lett. 123(2017), 107–113.
  DOI: https://doi.org/10.1016/j.spl.2016.12.006
• S. Bouzebda, On the weak convergence and the uniform-in-bandwidth consistency of the general conditional U-processes based on the copula representation: multivariate setting, Hacet. J. Math. Stat. 52 (2023), 1303–1348.
  DOI: https://doi.org/10.15672/hujms.1134334
• S. Bouzebda, N.-E. El Faouzi, New two-sample tests based on the integrated empirical copula processes, Statistics 46(2012), 313–324.
  DOI: https://doi.org/10.1080/02331888.2010.512976
• S. Bouzebda, A. A. Ferfache, Asymptotic properties of M-estimators based on estimating equations and censored data in semi-parametric models with multiple change points, J. Math. Anal. Appl. 497 (2021), Paper No. 124883, 44 pp.
  DOI: https://doi.org/10.1016/j.jmaa.2020.124883
• S. Bouzebda, A. A. Ferfache, Asymptotic properties of semiparametric M-estimators with multiple change points, Phys. A609 (2023), Paper No. 128363, 29 pp.
  DOI: https://doi.org/10.1016/j.physa.2022.128363
• S. Bouzebda, A. Keziou, T. Zari, K-sample problem using strong approximations of empirical copula processes, Math. Methods Statist. 20 (2011), 14–29.
  DOI: https://doi.org/10.3103/S1066530711010029
• B. E. Brodsky, B. S. Darkhovsky, Nonparametric Methods in Change-Point Problems, Mathematics and its Applications, Vol. 243, Kluwer Academic Publishers, Dordrecht, 1993, 209 pp.
  DOI: https://doi.org/10.1007/978-94-015-8163-9
• S. Bouzebda, T. Zari, Asymptotic behavior of weighted multivariate Cramér–von Mises-type statistics under contiguous alternatives, Math. Methods Statist. 22 (2013), 226–252.
  DOI: https://doi.org/10.3103/S1066530713030046
• M. D. Burke, M. Csörgő, S. Csörgő, P. Révész, Approximations of the empirical process when parameters are estimated, Ann. Probab. 7 (1979), 790–810.
  DOI: https://doi.org/10.1214/aop/1176994939
• J. Chan, L. Horváth, M. Hušková, Darling–Erdős limit results for change-point detection in panel data, J. Statist. Plann. Inference 143 (2013), 955–970.
  DOI: https://doi.org/10.1016/j.jspi.2012.11.004
• J. Chen, A. K. Gupta, Parametric Statistical Change Point Analysis, Birkhäuser, Boston, MA, 2000, 184 pp.
  DOI: https://doi.org/10.1007/978-1-4757-3131-6
• M. Csörgő, A glimpse of the KMT (1975) approximation of empirical processes by Brownian bridges via quantiles, Acta Sci. Math. (Szeged) 73 (2007), 349–366.
• M. Csörgő, L. Horváth, A note on strong approximations of multivariate empirical processes, Stochastic Process. Appl. 28(1988), 101–109.
  DOI: https://doi.org/10.1016/0304-4149(88)90068-3
• M. Csörgő, L. Horváth, Weighted Approximations in Probability and Statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Chichester, 1993, 442 pp.
• M. Csörgő, L. Horváth, Limit Theorems in Change-Point Analysis, Wiley Series in Probability and Statistics, John Wiley & Sons, Chichester, 1997, 414 pp.
• M. Csörgő, P. Révész, Strong Approximations in Probability and Statistics, Probability and Mathematical Statistics, Academic Press, New York–London, 1981, 284 pp.
• S. Csörgő, P. Hall, The Komlós–Major–Tusnády approximations and their applications, Austral. J. Statist. 26 (1984), 189–218.
  DOI: https://doi.org/10.1111/j.1467-842X.1984.tb01233.x
• A. DasGupta, Asymptotic Theory of Statistics and Probability, Springer Texts in Statistics, Springer, New York, 2008, 722 pp.
• J. Durbin, Weak convergence of the sample distribution function when parameters are estimated, Ann. Statist. 1 (1973), 279–290.
  DOI: https://doi.org/10.1214/aos/1176342365
• E. Giné, R. Latała, J. Zinn, Exponential and moment inequalities for U-statistics, in: High Dimensional Probability II, Progr. Probab., Vol. 47, Birkhäuser, Boston, MA, 2000, 13–38.
• N. Henze, Ya. Yu. Nikitin, A new approach to goodness-of-fit testing based on the integrated empirical process, J. Nonparametr. Statist. 12 (2000), 391–416.
  DOI: https://doi.org/10.1080/10485250008832815
• N. Henze, Ya. Yu. Nikitin, Watson-type goodness-of-fit tests based on the integrated empirical process, Math. Methods Statist. 11 (2002), 183–202.
• N. Henze, Ya. Yu. Nikitin, Two-sample tests based on the integrated empirical process, Comm. Statist. Theory Methods 32(2003), 1767–1788.
  DOI: https://doi.org/10.1081/STA-120022708
• L. Horváth, G. Rice, Extensions of some classical methods in change point analysis, TEST 23 (2014), 219–255.
  DOI: https://doi.org/10.1007/s11749-014-0368-4
• P. Jing, J. Wang, Testing the equality of multivariate distributions using the bootstrap and integrated empirical processes, Comm. Statist. Theory Methods 35 (2006), 661–670.
  DOI: https://doi.org/10.1080/03610920500498816
• P. Jing, Y. Yang, Testing the equality of multivariate distributions using the integrated empirical processes, Math. Appl. (Wuhan) 20 (2007), 614–620.
• J. Komlós, P. Major, G. Tusnády, An approximation of partial sums of independent RV’s and the sample DF. I, Z. Wahrscheinlichkeitstheorie verw. Gebiete 32 (1975), 111–131.
  DOI: https://doi.org/10.1007/BF00533093
• A. Lachal, Study of some new integrated statistics: computation of Bahadur efficiency, relation with non-standard boundary value problems, Math. Methods Statist. 10 (2001), 73–104.
• D. M. Mason, H. H. Zhou, Quantile coupling inequalities and their applications, Probab. Surv. 9 (2012), 439–479.
  DOI: https://doi.org/10.1214/12-PS198
• G. Neuhaus, On weak convergence of stochastic processes with multidimensional time parameter, Ann. Math. Statist. 42(1971), 1285–1295.
  DOI: https://doi.org/10.1214/aoms/1177693241
• F. H. Ruymgaart, Asymptotic normality of nonparametric tests for independence, Ann. Statist. 2 (1974), 892–910.
  DOI: https://doi.org/10.1214/aos/1176342812
• F. H. Ruymgaart, G. R. Shorack, W. R. van Zwet, Asymptotic normality of nonparametric tests for independence, Ann. Math. Statist. 43 (1972), 1122–1135.
  DOI: https://doi.org/10.1214/aoms/1177692465
• G. R. Shorack, J. A. Wellner, Empirical Processes with Applications to Statistics, Classics in Applied Mathematics, Vol. 59, SIAM, Philadelphia, PA, 2009, 956 pp.
  DOI: https://doi.org/10.1137/1.9780898719017
• B. Szyszkowicz, Asymptotic distributions of weighted pontograms under contiguous alternatives, Math. Proc. Cambridge Philos. Soc. 112 (1992), 431–447.
  DOI: https://doi.org/10.1017/S0305004100071097
• A. W. van der Vaart, J. A. Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics, 2nd ed., Springer Series in Statistics, Springer, Cham, 2023, 679 pp.
  DOI: https://doi.org/10.1007/978-3-031-29040-4

Submitted: April 2, 2026 / Accepted: June 6, 2026 / Published: September 28, 2026

Download PDF

How to cite: S. Bouzebda, Strong approximations for multivariate integrated empirical processes: Two-sample testing, change-point detection, and inference with estimated parameters,
Journal of Advanced Analysis and Applications, 1(2) (2026), 1–57.
DOI: https://doi.org/10.66745/jaaa.v1i2.001

This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). https://creativecommons.org/licenses/by/4.0/