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Real-time detection of a change-point in a linear expectile model

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Abstract

In the present paper we address the real-time detection problem of a change-point in the coefficients of a linear model with the possibility that the model errors are asymmetrical and that the explanatory variables number is large. We build test statistics based on the cumulative sum (CUSUM) of the expectile function derivatives calculated on the residuals obtained by the expectile and adaptive LASSO expectile estimation methods. The asymptotic distribution of these statistics are obtained under the hypothesis that the model does not change. Moreover, we prove that they diverge when the model changes at an unknown observation. The asymptotic study of the test statistics under these two hypotheses allows us to find the asymptotic critical region and the stopping time, that is the observation where the model will change. The empirical performance is investigated by a comparative simulation study with other statistics of CUSUM type. Two examples on real data are also presented to demonstrate its interest in practice.

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References

  • Cassotti M, Ballabio D, Todeschini R, Consonni V (2015) A similarity-based QSAR model for predicting acute toxicity towards the fathead minnow (Pimephales promelas). SAR QSAR Environ Res 26:217–243

    Article  Google Scholar 

  • Chen Z, Hu Y (2017) Cumulative sum estimator for change-point in panel data. Statist Papers 58(3):707–728

    Article  MathSciNet  Google Scholar 

  • Ciuperca G (2015) Real time change-point detection in a model by adaptive LASSO and CUSUM. J SFdS 156(4):113–132

    MathSciNet  MATH  Google Scholar 

  • Ciuperca G (2016) Adaptive LASSO model selection in a multiphase quantile regression. Statistics 50(5):1100–1131

    Article  MathSciNet  Google Scholar 

  • Ciuperca G (2017) Real time change-point detection in a nonlinear quantile model. Seq Anal 36(1):1–23

    Article  MathSciNet  Google Scholar 

  • Ciuperca G (2018) Test by adaptive LASSO quantile method for real-time detection of a change-point. Metrika 81:689–720

    Article  MathSciNet  Google Scholar 

  • Ciuperca G (2021) Variable selection in high-dimensional linear model with possibly asymmetric errors. Comput Statist Data Anal 155:107112

    Article  MathSciNet  Google Scholar 

  • Geng J, Zhang B, Huie LM, Lai L (2019) Online change-point detection of linear regression models. IEEE Trans Signal Process 67(12):3316–3329

    Article  MathSciNet  Google Scholar 

  • Götze F, Zaitsev AY (2009) Bounds for the rate of strong approximation in the multidimensional invariance principle. Theory Probab Appl 53(1):59–80

    Article  MathSciNet  Google Scholar 

  • Gronneberg S, Holcblat B (2019) On partial-sum processes of ARMAX residuals. Ann Stat 47(6):3216–3243

    Article  MathSciNet  Google Scholar 

  • Gu Y, Zou H (2016) High-dimensional generalizations of asymmetric least squares regression and their applications. Ann Stat 44(6):2661–2694

    Article  MathSciNet  Google Scholar 

  • Hastie T, Tibshirani R, Wainwright M (2015) Statistical learning with sparsity. The lasso and generalizations. Monographs on Statistics and Applied Probability 143. CRC Press, Boca Raton. xv+351 pp. ISBN: 978-1-4987-1216-3

  • Hoeffding W (1963) Probability inequalities for sums of bounded random variables. J Am Stat Assoc 58:13–30

    Article  MathSciNet  Google Scholar 

  • Horváth L, Rice G (2016) Asymptotics for empirical eigenvalue processes in high-dimensional linear factor models. J Multivar Anal 169:138–165

    Article  MathSciNet  Google Scholar 

  • Horváth L, Hušková M, Kokoszka P, Steinebach J (2004) Monitoring changes in linear models. J Stat Plan Inference 126(1):225–251

    Article  MathSciNet  Google Scholar 

  • Hušková M, Kirch C (2012) Bootstrapping sequential change-point tests for linear regression. Metrika 75(5):673–708

    Article  MathSciNet  Google Scholar 

  • Jiang P, Kurozumi E (2019) Power properties of the modified CUSUM tests. Commun Stat Theory Methods 48(12):2962–2981

    Article  MathSciNet  Google Scholar 

  • Jiang P, Kurozumi E (2020) Monitoring parameter changes in models with a trend. J Stat Plan Inference 207:288–319

    Article  MathSciNet  Google Scholar 

  • Kirch C, Weber S (2018) Modified sequential change point procedures based on estimating functions. Electron J Stat 12(1):1579–1613

    Article  MathSciNet  Google Scholar 

  • Koenker R (2005) Quantile regression econometric society monographs, vol 38. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Komlós J, Major P, Tusnády G (1975) An approximation of partial sums of independent RV’s, and the sample DF. I. Z Wahrsch Verw Gebiete 32:111–131

    Article  MathSciNet  Google Scholar 

  • Komlós J, Major P, Tusnády G (1976) An approximation of partial sums of independent RV’s, and the sample DF. II. Z Wahrsch Verw Gebiete 34:33–58

    Article  MathSciNet  Google Scholar 

  • Leisch F, Hornik K (2000) Monitoring structural changes with the generalized fluctuation test. Econ Theory 16:835–854

    Article  MathSciNet  Google Scholar 

  • Liao L, Park C, Choi H (2019) Penalized expectile regression: an alternative to penalized quantile regression. Ann Inst Stat Math 71(2):409–438

    Article  MathSciNet  Google Scholar 

  • Liu B, Zhou C, Zhang X (2019) A tail adaptive approach for change point detection. J Multivar Anal 169:33–48

    Article  MathSciNet  Google Scholar 

  • Nedényi FK (2018) An online change detection test for parametric discrete-time stochastic processes. Seq Anal 37(2):246–267

    Article  MathSciNet  Google Scholar 

  • Newey WK, Powell JL (1987) Asymmetric least squares estimation and testing. Econometrica 55(4):818–847

    Article  MathSciNet  Google Scholar 

  • Oh H, Lee S (2019) Modified residual CUSUM test for location-scale time series models with heteroscedasticity. Ann Inst Stat Math 71(5):1059–1091

    Article  MathSciNet  Google Scholar 

  • Prášková Z (2018) Change point detection in vector autoregression. Kybernetika 54(6):1122–1137

    MathSciNet  MATH  Google Scholar 

  • Qi P, Duan X, Tian Z, Li F (2016) Sequential monitoring for changes in models with a polynomial trend. Commun Stat Simul Comput 45(1):222–239

    Article  MathSciNet  Google Scholar 

  • Sahhanenko AI (1985) Estimates in an invariance principle. Trudy Inst Mat SO AN SSSR vol 5 Nauka Novosibirsk (Russian): 27–44

  • Song J, Kang J (2020) Sequential change point detection in ARMA-GARCH models. J Stat Comput Simul 90(8):1520–1538

    Article  MathSciNet  Google Scholar 

  • Zhang F, Li Q (2017) A continuous threshold expectile model. Comput Statist Data Anal 116:49–66

    Article  MathSciNet  Google Scholar 

  • Zhang F, Li Q (2017) Robust bent line regression. J Stat Plan Inference 185:41–55

    Article  MathSciNet  Google Scholar 

  • Zhao J, Chen Y, Zhang Y (2018) Expectile regression for analyzing heteroscedasticity in high dimension. Stat Probab Lett 137:304–311

    Article  MathSciNet  Google Scholar 

  • Zhou M, Wang HJ, Tang Y (2015) Sequential change point detection in linear quantile regression models. Stat Probab Lett 100:98–103

    Article  MathSciNet  Google Scholar 

  • Zou H (2006) The adaptive Lasso and its oracle properties. J Am Stat Assoc 101(476):1418–1428

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author sincerely thanks the two anonymous referees for their valuable and constructive suggestions which improved the quality of the paper.

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  1. Université Lyon 1, CNRS, UMR 5208, Institut Camille Jordan, Bat. Braconnier, 43, blvd du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Gabriela Ciuperca

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  1. Gabriela Ciuperca
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Correspondence to Gabriela Ciuperca.

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Ciuperca, G. Real-time detection of a change-point in a linear expectile model. Stat Papers 63, 1323–1367 (2022). https://doi.org/10.1007/s00362-021-01278-5

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