This paper is devoted to the study of some asymptotic properties of a -estimator in a framework of detection of abrupt changes in random field’s distribution. This class of problems includes e.g. recovery of sets. It involves various techniques, including -estimation method, concentration inequalities, maximal inequalities for dependent random variables and -mixing. Penalization of the criterion function when the size of the true model is unknown is performed. All the results apply under mild, discussed assumptions. Simple examples are provided.
Mots-clés : detection of change-points, $M$-estimation, penalized $M$-estimation, concentration inequalities, maximal inequalities, mixing
@article{PS_2002__6__189_0, author = {Chambaz, Antoine}, title = {Detecting abrupt changes in random fields}, journal = {ESAIM: Probability and Statistics}, pages = {189--209}, publisher = {EDP-Sciences}, volume = {6}, year = {2002}, doi = {10.1051/ps:2002011}, mrnumber = {1943147}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2002011/} }
Chambaz, Antoine. Detecting abrupt changes in random fields. ESAIM: Probability and Statistics, Volume 6 (2002), pp. 189-209. doi : 10.1051/ps:2002011. http://www.numdam.org/articles/10.1051/ps:2002011/
[1] A new look at the statistical model identification. IEEE Trans. Automat. Control AC-19 (1974) 716-723. System identification and time-series analysis. | MR | Zbl
,[2] Non-parametric estimation for the location of a change-point in an otherwise smooth hazard function under random censoring. Scand. J. Statist. 27 (2000) 501-519. | MR | Zbl
, and ,[3] Model selection with data-oriented penalty. J. Statist. Plann. Inference 77 (1999) 103-117. | MR | Zbl
, and , , and P Massart, Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 (1999) 301-413. |[5] Detection of abrupt changes: Theory and application. Prentice Hall Inc. (1993). | MR
and ,[6] Nonparametric methods in change-point problems. Kluwer Academic Publishers Group (1993). | MR | Zbl
and ,[7] Change-point problems. Institute of Mathematical Statistics, Hayward, CA (1994). Papers from the AMS-IMS-SIAM Summer Research Conference held at Mt. Holyoke College, South Hadley, MA July 11-16, 1992. | MR
, and ,[8] The estimation of the order of a mixture model. Bernoulli 3 (1997) 279-299. | Zbl
and ,[9] Exponential inequalities and functional central limit theorems for random fields. ESAIM P&S 5 (2001) 77. | Numdam | MR | Zbl
,[10] Mixing. Springer-Verlag, New York (1994). Properties and examples. | MR | Zbl
,[11] On the use of penalized contrasts for solving inverse problems. Application to the DDC (Detection of Divers Changes) problem (submitted).
,[12] Detection of multiple changes in a sequence of dependent variables. Stochastic Process. Appl. 83 (1999) 79-102. | MR | Zbl
,[13] An application of MCMC methods for the multiple change-points problem. Signal Process. 81 (2001) 39-53. | Zbl
and ,[14] The multiple change-points problem for the spectral distribution. Bernoulli 6 (2000) 845-869. | MR | Zbl
and ,[15] Least-squares estimation of an unknown number of shifts in a time series. J. Time Ser. Anal. 21 (2000) 33-59. | MR | Zbl
and ,[16] Lectures on statistical learning theory. Presented at the Garchy Seminar on Mathematical Statistics and Applications, available at http://www.econ.upf.es/~lugosi (2000).
,[17] Asymptotical minimax recovery of sets with smooth boundaries. Ann. Statist. 23 (1995) 502-524. | MR | Zbl
and ,[18] Some applications of concentration inequalities to statistics. Ann. Fac. Sci. Toulouse Math. (6) 9 (2000) 245-303. | Numdam | MR | Zbl
,[19] A general moment inequality for the maximum of the rectangular partial sums of multiple series. Acta Math. Hungar. 41 (1983) 337-346. | MR | Zbl
,[20] Moment and probability bounds with quasisuperadditive structure for the maximum partial sum. Ann. Probab. 10 (1982) 1032-1040. | MR | Zbl
, and ,[21] Limit theorems of probability theory. The Clarendon Press Oxford University Press, New York (1995). Sequences of independent random variables, Oxford Science Publications. | MR | Zbl
,[22] Théorie asymptotique des processus aléatoires faiblement dépendants. Springer (2000). | MR | Zbl
,[23] Estimating the dimension of a model. Ann. Statist. 6 (1978) 461-464. | MR | Zbl
,[24] Contributions to central limit theory for dependent variables. Ann. Math. Statist. 39 (1968) 1158-1175. | MR | Zbl
,[25] New concentration inequalities in product spaces. Invent. Math. 126 (1996) 505-563. | MR | Zbl
,[26] Asymptotic statistics. Cambridge University Press (1998). | MR | Zbl
,[27] Weak convergence and empirical processes. Springer-Verlag, New York (1996). With applications to statistics. | MR | Zbl
and ,[28] Statistical learning theory. John Wiley & Sons Inc., New York (1998). | MR | Zbl
,[29] Estimating the number of change-points via Schwarz's criterion. Statist. Probab. Lett. 6 (1988) 181-189. | Zbl
,Cited by Sources: