On Designs for Recursive Least Squares Residuals to Detect Alternatives

  • Wolfgang BischoffEmail author
Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


Linear regression models are checked by a lack-of-fit (LOF) test to be sure that the model is at least approximatively true. In many practical cases data are sampled sequentially. Such a situation appears in industrial production when goods are produced one after the other. So it is of some interest to check the regression model sequentially. This can be done by recursive least squares residuals. A sequential LOF test can be based on the recursive residual partial sum process. In this paper we state the limit of the partial sum process of a triangular array of recursive residuals given a constant regression model when the number of observations goes to infinity. Furthermore, we state the corresponding limit process for local alternatives. For specific alternatives designs are determined dominating other designs in respect of power of the sequential LOF test described above. In this context a result is given in which e−1 plays a crucial role.


  1. 1.
    Bischoff, W.: A functional central limit theorem for regression models. Ann. Stat. 26, 1398–1410 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bischoff, W., Gegg, A.: Partial sum process to check regression models with multiple correlated response: with an application for testing a change point in profile data. J. Multivar. Anal. 102, 281–291 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bischoff, W., Miller, F.: Asymptotically optimal tests and optimal designs for testing the mean in regression models with applications to change-point problems. Ann. Inst. Stat. Math. 52, 658–679 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bischoff, W., Somayasa, W.: The limit of the partial sums process of spatial least squares residuals. J. Multivar. Anal. 100, 2167–2177 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Brown, R.L., Durbin, J., Evans, J.M.: Techniques for testing the constancy of regression relationships over time. J. R. Stat. Soc. Ser. B 37, 149–192 (1975)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bruss, F.T.: A unified approach to a class of best choice problems with an unknown number of options. Ann. Probab. 12, 882–889 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Farebrother, R.W.: An historical note on recursive residuals. J. R. Stat. Soc. Ser. B 40, 373–375 (1978)MathSciNetGoogle Scholar
  8. 8.
    Gardner, L.A.: On detecting changes in the mean of normal variates. Ann. Math. Stat. 40, 116–126 (1969)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Jandhyala, V.K., MacNeill, I.B.: Residual partial sum limit process for regression models with applications to detecting parameter changes at unknown times. Stoch. Process. Appl. 33, 309–323 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Jandhyala, V.K., MacNeill, I.B.: Tests for parameter changes at unknown times in linear regression models. J. Stat. Plan. Inference 27, 291–316 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Jandhyala, V.K., MacNeill, I.B.: Iterated partial sum sequences of regression residuals and tests for changepoints with continuity constraints. J. R. Stat. Soc. Ser. B 59, 147–156 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Jandhyala, V.K., Zacks, S., El-Shaarawi, A.H.: Change–point methods and their applications: contributions of Ian MacNeill. Environmetrics 10, 657–676 (1999)CrossRefGoogle Scholar
  13. 13.
    MacNeill, I.B.: Properties of sequences of partial sums of polynomial regression residuals with applications to tests for change of regression at unknown times. Ann. Stat. 6, 422–433 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    MacNeill, I.B.: Limit processes for sequences of partial sums of regression residuals. Ann. Probab. 6, 695–698 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Rabovski, O.: Asymptotische Tests basierend auf rekursiven Residuen von Regressionsmodellen, Diplomarbeit (2003)Google Scholar
  16. 16.
    Sen, P.K.: Invariance principles for recursive residuals. Ann. Stat. 10, 307–312 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Sen, A., Srivastava, M.S.: On tests for detecting change in mean when variance is unknown. Ann. Inst. Stat. Math. 27, 479–486 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Shorack, G.R.: Probability for Statisticans. Springer, New York [u.a.] (2000)Google Scholar
  19. 19.
    Watson, G.S.: Detecting a change in the intercept in multiple regression. Stat. Probab. Lett. 23, 69–72 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Xie, L., MacNeill, I.B.: Spatial residual processes and boundary detection. S. Afr. Stat. J. 40, 33–53 (2006)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Mathematisch-Geographische FakultaetKatholische Universitaet Eichstaett-IngolstadtEichstaettGermany

Personalised recommendations