pp 1-7 | Cite as

A Bootstrap Approach to Testing for Time-Variability of AR Process Coefficients in Regression Time Series with t-Distributed White Noise Components

  • Hamza AlkhatibEmail author
  • Mohammad Omidalizarandi
  • Boris Kargoll
Part of the International Association of Geodesy Symposia book series


In this paper, we intend to test whether the random deviations of an observed regression time series with unknown regression coefficients can be described by a covariance-stationary autoregressive (AR) process, or whether an AR process with time-variable (say, linearly changing) coefficients should be set up. To account for possibly present multiple outliers, the white noise components of the AR process are assumed to follow a scaled (Student) t-distribution with unknown scale factor and degree of freedom. As a consequence of this distributional assumption and the nonlinearity of the estimator, the distribution of the test statistic is analytically intractable. To solve this challenging testing problem, we propose a Monte Carlo (MC) bootstrap approach, in which all unknown model parameters and their joint covariance matrix are estimated by an expectation maximization algorithm. We determine and analyze the power function of this bootstrap test via a closed-loop MC simulation. We also demonstrate the application of this test to a real accelerometer dataset within a vibration experiment, where the initial measurement phase is characterized by transient oscillations and modeled by a time-variable AR process.


Bootstrap test EM algorithm Monte Carlo simulation Regression time series Scaled t-distribution Time-variable autoregressive process 



Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—386369985. The presented application of the PCB Piezotronics accelerometer within the vibration analysis experiment was performed as a part of the collaborative project “Spatio-temporal monitoring of bridge structures using low cost sensors” with ALLSAT GmbH, which is funded by the German Federal Ministry for Economic Affairs and Energy (BMWi) and the Central Innovation Programme for SMEs (ZIM Kooperationsprojekt, ZF4081803DB6). In addition, the authors acknowledge the Institute of Concrete Construction (Leibniz Universität Hannover) for providing the shaker table and the reference accelerometer used within this experiment.


  1. Alkhatib H, Kargoll B, Paffenholz JA (2018) Further results on robust multivariate time series analysis in nonlinear models with autoregressive and t-distributed errors. In: Rojas I, Pomares H, Valenzuela O (eds) Time series analysis and forecasting. ITISE 2017. Contributions to statistics. Springer, Cham. pp 25–38. Scholar
  2. Angrisano A, Maratea A, Gaglione S (2018) A resampling strategy based on bootstrap to reduce the effect of large blunders in GPS absolute positioning. J Geod 92:81–92. Scholar
  3. Baarda W (1968) A testing procedure for use in geodetic networks. Publications on Geodesy (New Series), vol 2, no 5, Netherlands Geodetic Commission, DelftGoogle Scholar
  4. Davidson R, MacKinnon JG (2000) Bootstrap tests: How many bootstraps? Economet Rev 19:55–68. Scholar
  5. Efron B (1979) Bootstrap methods: another look at the jackknife. Ann Stat 7:1–26. Scholar
  6. Kargoll B, Omidalizarandi M, Loth I, Paffenholz JA, Alkhatib H (2018a) An iteratively reweighted least-squares approach to adaptive robust adjustment of parameters in linear regression models with autoregressive and t-distributed deviations. J Geod 92:271–297. Scholar
  7. Kargoll B, Omidalizarandi M, Alkhatib H, Schuh WD (2018b) Further results on a modified EM algorithm for parameter estimation in linear models with time-dependent autoregressive and t-distributed errors. In: Rojas I, Pomares H, Valenzuela O (eds) Time series analysis and forecasting. ITISE 2017. Contributions to statistics. Springer, Cham, pp 323–337. Scholar
  8. Koch KR (2018a) Bayesian statistics and Monte Carlo methods. J Geod Sci 8:18–29. Scholar
  9. Koch KR (2018b) Monte Carlo methods. Int J Geomath 9:177–143. Scholar
  10. Koch KR, Kargoll B (2013) Expectation-maximization algorithm for the variance-inflation model by applying the t-distribution. J Appl Geod 7:217–225. Scholar
  11. Kuhlmann H (2001) Importance of autocorrelation for parameter estimation in regression models. In: 10th FIG international symposium on deformation measurements, pp 354–361Google Scholar
  12. Kutterer H (1999) On the sensitivity of the results of least-squares adjustments concerning the stochastic model. J Geod 73(7):350–361. Scholar
  13. Li H, Maddala GS (1996) Bootstrapping time series models. Econom Rev 15:115–158. Scholar
  14. Lösler M, Eschelbach C, Haas R (2018) Bestimmung von Messunsicherheiten mittels Bootstrapping in der Formanalyse. Zeitschrift für Geodäsie, Geoinformatik und Landmanagement zfv 143:224–232Google Scholar
  15. McKinnon J (2007) Bootstrap hypothesis testing. Queen’s Economics Department Working Paper No. 1127, Queen’s University, Kingston, Ontario, CanadaGoogle Scholar
  16. McLachlan GJ, Krishnan T (2008) The EM algorithm and extensions, 2nd edn. Wiley, HobokenGoogle Scholar
  17. Neuner H, Wieser A, Krähenbühl N (2014) Bootstrapping: Moderne Werkzeuge für die Erstellung von Konfidenzintervallen. In: Neuner H (ed) Zeitabhängige Messgrößen - Ihre Daten haben (Mehr-)Wert. Schriftenreihe des DVW, vol 74, pp 151–170Google Scholar
  18. Parzen E (1979) A density-quantile function perspective on robust estimation. In: Launer L, Wilkinson GN (eds) Robustness in statistics. Academic, New York, pp 237–258. Scholar
  19. Politis DN (2003) The impact of bootstrap methods on time series analysis. Statist Sci 18:219–230. Scholar
  20. Schuh WD (2003) The processing of band-limited measurements; filtering techniques in the least squares context and in the presence of data gaps. Space Sci Rev 108:67–78. Scholar
  21. Schuh WD, Brockmann JM (2016) Refinement of the stochastic model for GOCE gravity gradients by non-stationary decorrelation filters. ESA Living Planet Symposium 2016, Prag, Poster 2382Google Scholar
  22. Teunissen (1998) Success probability of integer GPS ambiguity rounding and bootstrapping. J Geod 72:606–612. Scholar
  23. Wiśniewski Z (2014) M-estimation with probabilistic models of geodetic observations. J Geod 88(10):941–957. Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Hamza Alkhatib
    • 1
    Email author
  • Mohammad Omidalizarandi
    • 1
  • Boris Kargoll
    • 2
  1. 1.Geodetic InstituteLeibniz University HannoverHannoverGermany
  2. 2.Institut für Geoinformation und Vermessung DessauAnhalt University of Applied SciencesDessau-RoßlauGermany

Personalised recommendations