Change point estimation in regression models with fixed design. (English) Zbl 1407.62104
Rykov, Vladimir V. (ed.) et al., Mathematical and statistical models and methods in reliability. Applications to medicine, finance, and quality control. Invited papers based on the presentation at the 6th international conference (MMR 2009), Moscow, Russia, June 22–26, 2009. Boston, MA: Birkhäuser. Stat. Ind. Technol., 207-221 (2010).
Summary: In this paper, we consider a simple regression model with change points in the regression function which can be one of two types: A so called smooth bent-line change point or a discontinuity point of a regression function. In both cases we investigate the consistency of the M-estimates of the change points. It turns out that the rates of convergence are \(n^{1/2}\) or \(n\), respectively, where \(n\) denotes the sample size in a fixed design. In addition, the asymptotic distributions of the change point estimators are investigated.
For the entire collection see [Zbl 1203.60007].
For the entire collection see [Zbl 1203.60007].
MSC:
| 62G05 | Nonparametric estimation |
| 62G08 | Nonparametric regression and quantile regression |
| 62G20 | Asymptotic properties of nonparametric inference |
Software:
timeregReferences:
| [1] | Aalen, O.O., Borgan, O., Gjessing, S.: Survival and Event History: A Process Point of View. Springer, New York (2008) · Zbl 1204.62165 · doi:10.1007/978-0-387-68560-1 |
| [2] | Andrews, D.W.K.: Consistency in nonlinear econometric models: A generic uniform law of large numbers. Econometrica, 55, 1465-1471 (1987) · Zbl 0646.62101 · doi:10.2307/1913568 |
| [3] | Bai, J.: Estimation of a change point in multiple regression models. Review of Economics and Statistics, 79, 551-560 (1997) · doi:10.1162/003465397557132 |
| [4] | Chow, Y.S., Teicher, H.: Probability Theorey. Springer, New York (1988) · Zbl 0652.60001 · doi:10.1007/978-1-4684-0504-0 |
| [5] | Dempfle, A., Stute, W.: Nonparametric estimation of a discontinuity in regression. Statistica Neerlandica, 56, 233-242 (2002) · Zbl 1076.62520 · doi:10.1111/1467-9574.00196 |
| [6] | Ferger, D.: Exponential and polynomial tailbounds for change-point estimators. Journal of Statistical Planning and Inference, 92, 73-109 (2001) · Zbl 0997.62018 · doi:10.1016/S0378-3758(00)00148-8 |
| [7] | Ferger, D.: A continuous mapping theorem for the argmax-functional in the non-unique case. Statistica Neerlandica, 58, 83-96 (2004) · Zbl 1090.60032 · doi:10.1046/j.0039-0402.2003.00111.x |
| [8] | Ferger, D.: Stochastische Prozesse mit Strukturbrüchen. Manuskript [in German], Dresdner Schriften zur Mathematischen Stochastik (2009) |
| [9] | Jensen, U., Lütkebohmert C.: Change-Point Models. Review in Ruggeri, F., Kenett, R., Faltin, W. (ed) Encyclopedia of Statistics in Quality and Reliability, Vol. 1, 306-312. Wiley, New York (2007) |
| [10] | Jensen, U., Lütkebohmert C.: A Cox-type regression model with change-points in the covariates. Lifetime Data Analysis, 14, 267-285 (2008) · Zbl 1356.62196 · doi:10.1007/s10985-008-9083-3 |
| [11] | Kosorok, M.R.: Introduction to Empirical Processes and Semiparametric Inference. Springer, New York (2008) · Zbl 1180.62137 · doi:10.1007/978-0-387-74978-5 |
| [12] | Kosorok, M., Song, R.: Inference under right censoring for transformation models with a change-point based on a covariate threshold. The Annals of Statistics, 35, 957-989 (2007) · Zbl 1136.62376 · doi:10.1214/009053606000001244 |
| [13] | Koul, H.L., Qian, L., Surgailis, D.: Asymptotics of M-estimators in two-phase linear regression models. Stochastic Process and their Applications, 103, 123-154 (2003) · Zbl 1075.62021 · doi:10.1016/S0304-4149(02)00185-0 |
| [14] | Martinussen, T., Scheike, T.H.: Dynamic Regression Models for Survival Data. Springer, New York (2006) · Zbl 1096.62119 |
| [15] | Lan, Y., Banerjee, M., Michailidis: Change-point estimation under adaptive sampling. The Annals of Statistics, 37, 1752-1791 (2009) · Zbl 1168.62018 · doi:10.1214/08-AOS602 |
| [16] | Müller, H.G.: Change-point in nonparametric regression analysis. The Annals of Statistics, 20, 737-761 (1992) · Zbl 0783.62032 · doi:10.1214/aos/1176348654 |
| [17] | Müller, H.G., Song, K.S.: Two-stage change-point estimators in smooth regression models. Statistics and Probability Letters, 34, 323-335 (1997) · Zbl 0874.62035 · doi:10.1016/S0167-7152(96)00197-6 |
| [18] | Müller, H.G., Stadtmüller, U.: Discontinuous versus smooth regression. The Annals of Statistics, 27, 299-337 (1999) · Zbl 0978.62056 · doi:10.1214/aos/1018031100 |
| [19] | Pons, O.: Estimation in a Cox regression model with a change-point according to a threshold in a covariate. The Annals of Statistics, 31, 442-463 (2003) · Zbl 1040.62090 · doi:10.1214/aos/1051027876 |
| [20] | Van der Vaart, A.W.: Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics (1998) · Zbl 0910.62001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
