Abstract
The existence of a discontinuity in a regression function can be inferred by comparing regression estimates based on the data lying on different sides of a point of interest. This idea has been used in earlier research by Hall and Titterington (1992), Müller (1992) and later authors. The use of nonparametric regression allows this to be done without assuming linear or other parametric forms for the continuous part of the underlying regression function. The focus of the present paper is on assessing the evidence for the presence of a discontinuity within a regression function through examination of the standardised differences of ‘left’ and ‘right’ estimators at a variety of covariate values. The calculations for the test are carried out through distributional results on quadratic forms. A graphical method in the form of a reference band to highlight the sources of the evidence for discontinuities is proposed. The methods are also developed for the two covariate case where there are additional issues associated with the presence of a jump location curve. Methods for estimating this curve are also developed. All the techniques, for the one and two covariate situations, are illustrated through applications.
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References
Bock M., Bowman A.W., and Ismail B. 2007. Estimation and inference for error variance in bivariate nonparametric regression. Statistics and Computing: to appear.
Bowman A.W. and Azzalini A. 1997. Applied Smoothing Techniques for Data Analysis. London, Oxford University Press.
Bowman A.W. and Azzalini A. 2003. Computational aspects of nonparametric smoothing with illustrations from the sm library. Computational Statistics and Data Analysis 42: 545–560.
Carlstein E., Müller H.-G., and Siegmund D. 1994. Change-point problems. IMS, Hayward, CA.
Cobb G. 1978. The problem of the Nile: conditional solution to a changepoint problem. Biometrika 65: 243–251.
Delgado M.A and Hidalgo J. 2000. Nonparametric inference on structural breaks. J. Econometrics 96: 113–144.
Fan J. and Gijbels I. 1996. Local polynomial modelling and its applications. London, Chapman & Hall.
Gasser T., Sroka L., and Jennen-Steinmetz C. 1986. Residual variance and residual pattern in nonlinear regression. Biometrika 73: 625–33.
Hall P., Peng L., and Rau C. 2001. Local likelihood tracking of fault lines and boundaries. J. Roy. Statist. Soc., Series B 63: 569–582.
Hall P. and Rau C. 2000. Tracking a smooth fault line in a response surface. Ann. Statist. 28: 713–733.
Hall P. and Titterington D.M. 1992. Edge-preserving and peak-preserving smoothing. Technometrics 34: 429–440.
Herrmann E. 2000. Variance estimation and bandwidth selection for kernel regression. In Schimek M. (Ed.), Smoothing and Regression: Approaches, Computation and Application. New York, Wiley.
Imhof J.P. 1961. Computing the distribution of quadratic forms in normal variables. Biometrika48: 419–426.
Jose C.T. and Ismail B. 2001. Nonparametric inference in jump regression surface. J. Nonpar. Stat.13: 791–813.
Johnson N.L. and Kotz S. 1972. Distributions in Statistics: Continuous Univari-ate Distributions, Vol. II. New York, Wiley.
Korostelev A.P. and Tsybakov A.B. 1993. Minimax Theory of Image Reconstruction. Lecture Notes in Statistics, 82. New York, Springer.
Loader C. 1996. Change points estimation using nonparametric regression. Ann. Statist.24: 1667–1678.
Müller H.G. 1992. Change-points in nonparametric regression analysis. Ann. Statist. 20: 737–761.
Müller H.G. and Song K. 1994. Maximin estimation of multidimensional boundaries. J. Mult. Anal.50: 265–281.
Müller H.-G. and Stadtmüller U. 1999. Discontinuous versus smooth regression. Ann. Statist.27: 299–337.
Poiner I.R., Blaber S.J.M., Brewer D.T., Burridge C.Y., Caesar d., Connell M., Dennis D., Dews G.D., Ellis A.N., Farmer M., Fry G.J., Glaister J., Gribble N., Hill B.J., Long B.G., Milton D.A., Pitcher C.R., Proh D., Salini J.P., Thomas M.R., Toscas P., Veronise S., Wang Y.G., and Wassenberg T.J. 1997. The effects of prawn trawling in the far northern section of the Great Barrier Reef. Final report to GBRMPA and FRDC on 1991–96 research. CSIRO Division of Marine Research. Queensland Dept. of Primary Industries.
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.
Qiu P. and Yandell B. 1998. A local polynomial jump detection algorithm in nonparametric regression. Technometrics40: 141–152.
Qiu P. 2005. Image Processing and jump Regression Analysis. New York, Wiley.
Rice J. 1984. Bandwidth choice for nonparametric regression. Ann. Statist. 12: 1215–1230.
Ronn B. 2001. Nonparametric m.l.e. for shifted curves. J. R. Statist. Soc., Series B63: 243–259.
Rudemo M. and Stryhn H. 1994. Boundary estimation for star-shaped objects, In: Carlstein E., Muller H.G., and Siegmund D. (Eds.), Change-point Problems IMS Lecture Notes and Monographs Series, pp. 276–283.
Ruppert D. and Wand M.P. 1994. Multivariate locally weighted least squares regression. Annals of Statistics 22: 1346–1370
Simonoff J.S. 1996. Smoothing Methods in Statistics. New York, Springer-Verlag.
Tsybakov A.B. 1994. Multidimensional change point problems and boundary estimation, In: Carlstein E., Muller H.G., and Siegmund D. (Eds.), Change-point Problems IMS Lecture Notes and Monographs Series, pp. 317–329.
Wand M.P. and Jones M.C. 1994. Kernel Smoothing. New York, Chapman & Hall.
Wang Y. 1998. Change curve estimation via wavelets. J. Amer. Statist. Assoc. 93: 163–172.
Yap C. 2004. Detecting discontinuities using nonparametric smoothing techniques in correlated data. Ph.D. thesis, University of Glasgow.
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Bowman, A.W., Pope, A. & Ismail, B. Detecting discontinuities in nonparametric regression curves and surfaces. Stat Comput 16, 377–390 (2006). https://doi.org/10.1007/s11222-006-9618-y
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DOI: https://doi.org/10.1007/s11222-006-9618-y