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Smooth change point estimation in regression models with random design. (English) Zbl 1440.62128
Summary: We consider the problem of estimating the location of a change point $$\theta_0$$ in a regression model. Most change point models studied so far were based on regression functions with a jump. However, we focus on regression functions, which are continuous at $$\theta_0$$. The degree of smoothness $$q_0$$ has to be estimated as well. We investigate the consistency with increasing sample size $$n$$ of the least squares estimates $$(\hat{\theta}_n,\hat{q}_n)$$ of $$(\theta_0, q_0)$$. It turns out that the rates of convergence of $$\hat{\theta}_n$$ depend on $$q_0$$: for $$q_0$$ greater than $$1/2$$ we have a rate of $$\sqrt{n}$$ and the asymptotic normality property; for $$q_0$$ less than $$1/2$$ the rate is $$n^{1/(2q_0+1)}$$ and the change point estimator converges to a maximizer of a Gaussian process; for $$q_0$$ equal to $$1/2$$ the rate is $$\sqrt{n \cdot \ln (n)}$$. Interestingly, in the last case the limiting distribution is also normal.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62F12 Asymptotic properties of parametric estimators 62E20 Asymptotic distribution theory in statistics 62G20 Asymptotic properties of nonparametric inference
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##### References:
  Aue, A., Steinebach, J. (2002). A note on estimating the change-point of a gradually changing stochastic process. Statistics & Probability Letters, 56, 177-191. · Zbl 1065.62149  Bai, J. (1997). Estimation of a change point in multiple regression models. Review of Economics and Statistics, 79, 551-560.  Csörgö, M., Horváth, L. (1997). Limit Theorems in Change-Point Analysis. New York: Wiley. · Zbl 0884.62023  Dempfle, A., Stute, W. (2002). Nonparametric estimation of a discontinuity in regression. Statistica Neerlandica, 56, 233-242. · Zbl 1076.62520  Döring, M., Jensen, U. (2010). Change point estimation in regression models with fixed design. In V. Rykov, M. Nikulin, N. Balakrishnan (Eds.), Mathematical and Statistical Models and Methods in Reliability (pp. 207-222). New York: Springer.  Dufner, J., Jensen, U., Schumacher, E. (2004). Statistik mit SAS (in German). Wiesbaden: Teubner. · Zbl 1136.62309  Feder, P. L. (1975). On asymptotic distribution theory in segmented regression problems. The Annals of Statistics, $$3$$, 49-83. · Zbl 0324.62014  Gallant, A. R. (1987). Nonlinear Statistical Models. New York: Wiley. · Zbl 0611.62071  Hinkley, D. (1971). Inference in two-phase regression. Journal of the American Statistical Association, 66, 736-743. · Zbl 0226.62068  Hušková, M. (1999). Gradual changes versus abrupt changes. Journal of Statistical Planning and Inference, 76, 109-125. · Zbl 1054.62520  Hušková, M. (2001). A note on estimators of gradual changes. In M. de Gunst, C. Klaassen, A. van der Vaart (Eds.), State of the art in probability and statistics (Leiden, 1999), 36 (pp. 345-358). Institute of Mathematical Statistics Lecture Notes-Monograph Series. Beachwood: Institute of Mathematical Statistics.  Ibragimov, I. A., Has’minskii, R. Z. (1981). Statistical Estimation— Asymptotic Theory. New York: Springer.  Kim, J., Pollard, D. (1990). Cube root asymptotics. The Annals of Statistics, 18, 191-219. · Zbl 0703.62063  Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. New York: Springer. · Zbl 1180.62137  Koul, H. L., Qian, L., Surgailis, D. (2003). Asymptotics of M-estimators in two-phase linear regression models. Stochastic Process and their Applications, 103, 123-154. · Zbl 1075.62021  Lan, Y., Banerjee, M., Michailidis, G. (2009). Change-point estimation under adaptive sampling. The Annals of Statistics, 37, 1752-1791. · Zbl 1168.62018  Müller, H. G. (1992). Change-points in nonparametric regression analysis. The Annals of Statistics, 20, 737-761. · Zbl 0783.62032  Müller, H. G., Song, K. S. (1997). Two-stage change-point estimators in smooth regression models. Statistics & Probability Letters, 34, 323-335. · Zbl 0874.62035  Müller, H. G., Stadtmüller, U. (1999). Discontinuous versus smooth regression. The Annals of Statistics, 27, 299-337. · Zbl 0954.62052  Rukhin, A. L., Vajda, I. (1997). Change-Point Estimation as a Nonlinear Regression Problem. Statistics, 30, 181-200. · Zbl 0915.62013  Van de Geer, S. (2000). Empirical Processes in M-Estimation. Cambridge: Cambridge University Press. · Zbl 1179.62073  Van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge: Cambridge University Press. · Zbl 0910.62001  Van der Vaart, A. W., Wellner, J. A. (1996). Weak Convergence and Empirical Processes. New York: Springer. · Zbl 0862.60002  Wang, Y. (1995). Jump and sharp cusp detection by wavelets. Biometrika, 82, 385-397. · Zbl 0824.62031
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