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Analysis of a change-point regression problem in quality control by partial sums processes and Kolmogorov type tests

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Abstract

In practice, it is an important problem (especially in quality control) to secure that a known regression function occurs during a certain period in time. In the present paper, we consider the change-point problem that under the null hypothesis this known regression function occurs. As alternative, we consider a certain non-parametric class of functions that is of particular interest in quality control. We analyze this test problem by using partial sums of the data. Asymptotically, we get Brownian motion and Brownian motion with trend (≠0) under the hypothesis and under the alternative, respectively. We prove that tests based on partial sums have a larger power when the partial sums are taken from the time reversed data. This can be quantitatively determined in an asymptotic way by some new results on Kolmogorov type tests for Brownian motion with trend. We illustrate our results by a certain model that is interesting in quality control and by an example with real data.

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Author information

Correspondence to Wolfgang Bischoff.

Additional information

Supported in part by the Deutsche Forschungsgemeinschaft Grant Bi655.

Supported in part by the Deutsche Forschungsgemeinschaft Grant Bi655 and by the Swiss National Science Foundation Grant 20-55586.98.

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Bischoff, W., Hashorva, E., Hüsler, J. et al. Analysis of a change-point regression problem in quality control by partial sums processes and Kolmogorov type tests. Metrika 62, 85–98 (2005). https://doi.org/10.1007/s001840400354

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AMS 2000 subject classifications

  • Primary 60G70
  • 62G10
  • 62G32
  • 62M02
  • secondary 60F10
  • 60G15
  • 62J05

Keywords

  • Change-point problem
  • Quality control
  • Regression models
  • Partial sums processes
  • Signal-plus-noise model
  • Brownian motion with trend
  • Tests of Kolmogorov type
  • Extreme values
  • Large deviations