The Annals of Statistics

Sequential methods for design-adaptive estimation of discontinuities in regression curves and surfaces

Peter Hall and Ilya Molchanov

Full-text: Open access


In fault-line estimation in spatial problems it is sometimes possible to choose design points sequentially, by working one's way gradually through the "response plane," rather than distributing design points across the plane prior to conducting statistical analysis. For example, when estimating a change line in the concentration of resources on or under the sea bed, individual measurements can be particularly expensive to make. In such cases, sequential, design-adaptive methods are attractive. Appropriate methodology is largely lacking, however, and the potential advantages of taking a sequential approach are unclear. In the present paper we address both these problems. We suggest a methodology based on "sequential refinement with reassessment" that relies upon assessing the correctness of each sequential result, and reappraising previous results if significance tests show that there is reason for concern. We focus part of our attention on univariate problems, and we show how methods for the spatial case can be constructed from univariate ones.

Article information

Ann. Statist., Volume 31, Number 3 (2003), 921-941.

First available in Project Euclid: 25 June 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62L12: Sequential estimation
Secondary: 62G20: Asymptotic properties 62H11: Directional data; spatial statistics

Changepoint fault line hypothesis test nonparametric estimation recursive search methods spatial statistics


Hall, Peter; Molchanov, Ilya. Sequential methods for design-adaptive estimation of discontinuities in regression curves and surfaces. Ann. Statist. 31 (2003), no. 3, 921--941. doi:10.1214/aos/1056562467.

Export citation


  • CARLSTEIN, E., MÜLLER, H.-G. and SIEGMUND, D. (1994), eds. Change-Point Problems. IMS, Hay ward, CA.
  • CRESSIE, N. A. C. (1993). Statistics for Spatial Data, rev. ed. Wiley, New York.
  • GHOSH, M., MUKHOPADHy AY, N. and SEN, P. K. (1997). Sequential Estimation. Wiley, New York.
  • GIJBELS, I., HALL, P. and KNEÏP, A. (1999). On the estimation of jump points in smooth curves. Ann. Inst. Statist. Math. 51 231-251.
  • HALL, P. and RAIMONDO, M. (1997). Approximating a line thrown at random onto a grid. Ann. Appl. Probab. 7 648-665.
  • HALL, P. and RAIMONDO, M. (1998). On global performance of approximations to smooth curves using gridded data. Ann. Statist. 26 2206-2217.
  • HALL, P. and RAU, C. (2000). Tracking a smooth fault line in a response surface. Ann. Statist. 28 713-733.
  • KOROSTELEV, A. P. and TSy BAKOV, A. B. (1993). Minimax Theory of Image Reconstruction. Lecture Notes in Statist. 82. Springer, Berlin.
  • LOADER, C. L. (1996). Change-point estimation using nonparametric regression. Ann. Statist. 24 1667-1678.
  • MAMMEN, E. and TSy BAKOV, A. B. (1995). Asy mptotical minimax recovery of sets with smooth boundaries. Ann. Statist. 23 502-524.
  • MÜLLER, H.-G. and SONG, K.-S. (1997). Two-stage change-point estimators in smooth regression models. Statist. Probab. Lett. 34 323-335.
  • PRONZATO, L., Wy NN, H. P. and ZHIGLJAVSKY, A. A. (2000). Dy namical Search. Applications of Dy namical Sy stems in Search and Optimization. Chapman and Hall, London.
  • QIU, P. (1998). Discontinuous regression surfaces fitting. Ann. Statist. 26 2218-2245.
  • QIU, P. and YANDELL, B. (1997). Jump detection in regression surfaces. J. Comput. Graph. Statist. 6 332-354.
  • RAIMONDO, M. (1996). Modèles en rupture, situations non ergodique et utilisation de méthode d'ondelette. Ph.D. dissertation, Univ. Paris VII.
  • RUDEMO, M. and STRy HN, H. (1994). Approximating the distribution of maximum likelihood contour estimators in two-region images. Scand. J. Statist. 21 41-55.
  • RUPPERT, D. (1991). Stochastic approximation. In Handbook of Sequential Analy sis (B. K. Ghosh and P. K. Sen, eds.) 503-529. Dekker, New York.
  • SHORACK, G. R. and WELLNER, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • TITTERINGTON, D. M. (1985a). Common structure of smoothing techniques in statistics. Internat. Statist. Rev. 53 141-170.
  • TITTERINGTON, D. M. (1985b). General structure of regularization procedures in image reconstruction. Astronom. and Astrophy s. 144 381-387.
  • WANG, Y. (1995). Jump and sharp cusp detection by wavelets. Biometrika 82 385-397.
  • ZHIGLJAVSKY, A. A. (1991). Theory of Global Random Search. Kluwer, Dordrecht.