Discontinuous versus smooth regression

Annals of Statistics

Volumn 27, Issue 1, 1999, Pages 299-337

  Müller, Hans Georg ^a   Stadtmüller, Ulrich ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF ULM   (Germany)

Author keywords

Asymptotic linear model; Goodness of fit; Jump detection; Model selection; Rate of convergence; Saltatory growth; Test for discontinuity; U statistics; Variance estimation

Indexed keywords

EID: 0033243872     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1018031113     Document Type: Article

References (33)

1. BHATTACHARYA, P. K. and BROCKWELL, P. J. (1976). The minimum of an additive process with applications to signal estimation and storage theory. Z. Wahrsch. Verw. Gebiete 37 51-75.
2. BILLINGSLEY, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
3. BRAUN, J. V. and MÜLLER, H. G. (1998). Statistical methods for the segmentation of DNA sequences. Statist. Sci. 13 142-162.
4. CARLSTEIN, E., MÜLLER, H. G. and SIEGMUND, D. eds. (1994). Change-Point Problems. IMS, Hayward, CA.
5. CHURCHILL, G. A. (1992). Hidden Markov chains and the analysis of genome structure. Computers in Chemistry 6 107-115.
6. COBB, G. W. (1978). The problem of the Nile: conditional solution to a change-point problem. Biometrika 62 243-251.
7. DONOHO, D. L., JOHNSTONE, I. M., KERKYACHARIAN, G. and PICARD, D. (1995). Wavelet shrinkage: asymptopia? J. Roy. Statist. Soc. B 57 301-369.
8. DUBOWIK, C. (1996). Entdeckung von Changepoints und gewichtete U-Statistiken. Diploma thesis, Univ. Ulm.
9. EUBANK, R. L. and SPECKMAN, P. L. (1994). Nonparametric estimation of functions with jump discontinuities. In Change-Point Problems (E. Carlstein, H. G. Müller and D. Siegmund, eds.) IMS, 130-144. Hayward, CA.
10. FAN, J. and GIJBELS, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
11. FREDKIN, D. R. and RICE, J. (1992). Bayesian restoration of single-channel patch clamp recordings. Biometrika 48 427-448.
12. FRIEDMAN, J. and TIBSHIRANI, R. (1984). The monotone smoothing of scatterplots. Technometrics 26 243-250.
13. GASSER, T., SROKA, L. and JENNEN-STEINMETZ, C. (1986). Residual variance and residual pattern in nonlinear regression. Biometrika 73 625-633.
14. HALL, P., KAY, J. W. and TITTERINGTON, D. M. (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 521-528.
15. HALL, P. and TITTERINGTON, D. M. (1992). Edge-preserving and peak-preserving smoothing. Technometrics 34 429-440.
16. HEINRICHS, C., MUNSON, P. D., COUNTS, D. R., CUTLER, G. B. and BARON, J. (1995). Patterns of human growth. Science 268 442-445.
17. HINKLEY, D. V. (1970). Inference about the change-point in a sequence of random variables. Biometrika 57 1-16.
18. JARRETT, R. G. (1979). Time intervals between coal mining disasters. Biometrika 66 191-193.
19. LAMPL, M., CAMERON, N., VELDHUIS, J. D. and JOHNSON, M. L. (1995). Patterns of human growth. Science 268 445-447.
20. LAMPL, M., VELDHUIS, J. D. and JOHNSON, M. L. (1992). Saltation and stasis: A model of human growth. Science 258 801-803.
21. LEE, A. J. (1990). U-Statistics. Dekker, New York.
22. LOADER, C. (1996). Change-point estimation using nonparametric regression. Ann. Statist. 24 1667-1678.
23. MCELROY, F. W. (1967). A necessary and sufficient condition that the ordinary least-squares estimator be best linear unbiased. J. Amer. Statist. Assoc. 62 1302-1304.
24. MÜLLER, H. G. (1985). Empirical bandwidth choice for nonparametric kernel regression by means of pilot estimators. Statist. Decisions 2 193-206.
25. MÜLLER, H. G. (1992). Change-points in nonparametric regression analysis. Ann. Statist. 20 737-761.
26. MÜLLER, H. G. (1993). Comment on "Local regression: Automatic kernel carpentry," by T. Hastie and C. Loader. Statist. Sci. 8 134-139.
27. MÜLLER, H. G. and SONG, K. S. (1997). Two-stage change-point estimators in smooth regression models. Statist. Probab. Lett. 34 323-335.
28. RICE, J. (1984). Bandwidth choice for nonparametric kernel regression. Ann. Statist. 12 1215-1230.
29. SERFLING, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
30. WANG, Y. (1995). Jump and sharp cusp detection by wavelets. Biometrika 82 385-397.
31. WU, J. S. and CHU, C. K. (1993). Kernel-type estimators of jump points and values of a regression function. Ann. Statist. 21 1545-1566.
32. YAO, Y.-C. (1984). Estimation of a noisy discrete-time step function: Bayes and empirical Bayes approaches. Ann. Statist. 12 1117-1123.
33. YAO, Y.-C. and AU, S. T. (1989). Least-squares estimation of a step function. Sankhyā Ser. A 51 370-381.