A note on estimating a smooth monotone regression by combining kernel and density estimates

Journal of Nonparametric Statistics

Volumn 20, Issue 8, 2008, Pages 679-691

  Birke, Melanie ^a   Dette, Holger ^a  

^a RUHR UNIVERSITY BOCHUM   (Germany)

Author keywords

Greatest convex minorant; Monotone estimation; Nadaraya Watson estimate; Non decreasing rearrangements; Order restricted inference

Indexed keywords

EID: 55849141314     PISSN: 10485252     EISSN: 10290311     Source Type: Journal    
DOI: 10.1080/10485250802445399     Document Type: Article

References (26)

1. R.L. Matzkin, Restrictions of econometric theory in nonparametric methods, R.F. Engle and D.L. McFadden, eds., Elsevier, Amsterdam, 1994.
2. Y. Aït-Sahalia and J. Duarte, Nonparametric option pricing under shape restrictions, J. Econom. 116 (2003), pp. 9-17.
3. H.D. Brunk, Maximum likelihood estimates of monotone parameters, Ann. Math. Statist. 26 (1955), pp. 607-616.
4. K.F. Cheng and P.E. Lin, Nonparametric estimation of a regression function, Z. Wahrsch. Verw. Gebiete 57 (1981), pp. 223-233.
5. F.T. Wright, The asymptotic behavior of monotone regression estimates, Ann. Statist. 9 (1982), pp. 443-448.
6. H. Mukerjee, Monotone nonparametric regression, Ann. Statist. 16 (1988), pp. 741-750.
7. E. Mammen, Estimating a smooth monotone regression function, Ann. Statist. 19 (1991), pp. 724-740.
8. J. Friedman and R. Tibshirani, The monotone smoothing of scatterplots, Technometrics 26 (1984), pp. 243-250.
9. J.O. Ramsay, Monotone regression splines in action (with comments), Statist. Sci. 3 (1988), pp. 425-461.
10. C. Kelly and J. Rice, Monotone smoothing with application to dose response curves and the assessment of synergism, Biometrics 46 (1990), pp. 1071-1085.
11. E. Mammen and C. Thomas-Agnan, Smoothing splines and shape restrictions, Scand. J. Statist. 26 (1999), pp. 239-252.
12. E. Mammen, J.S. Marron, B.A. Turlach, and M.P. Wand, A general projection framework for constrained smoothing, Statist. Sci. 16 (2001), pp. 232-248.
13. P. Hall and L.-S. Huang, Nonparametric kernel regression subject to monotonicity constraints, Ann. Statist. 29 (2001), pp. 624-647.
14. M. Delecroix and C. Thomas-Agnan, Spline and kernel regression under shape restrictions, Smoothing and Regression. Approaches, Computation and Application, M.G. Schimek, ed., Wiley Series in Probability and Statistics, Wiley, Chichester, 2000, pp. 109-133.
15. I. Gijbels, Monotone regression, N. Balakrishnan, S. Kotz, C.B. Read, and B. Vadakovic, eds., Wiley, Hoboken, NJ, 2005.
16. H. Dette, N. Neumeyer, and K.F. Pilz, A simple nonparametric estimator of a strictly monotone regression function, Bernoulli 12 (2006), pp. 469-490.
17. G.H. Hardy, J.E. Littlewood, and G. Pólya, Inequalities, M.G. Schimek, ed., Cambridge University Press, Cambridge, 1952.
18. G.G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly 60 (1953), pp. 176-179.
19. T. Robertson, F.T. Wright, and R.L. Dykstra, Order Restricted Statistical Inference, Wiley Series in Probability and Mathematical Statistics, Wiley, Chichester, 1988.
20. M. Birke and H. Dette, Testing strict monotonicity in nonparametric regression, Math. Methods Statist. 16 (2007), pp. 1-13.
21. R.E. Barlow, D.J. Bartholomew, J.M. Bremner, and H.D. Brunk, Statistical Inference Under Order Restrictions, Wiley, New York, 1972.
22. J. Rice, Bandwidth choice for nonparametric regression, Ann. Statist. 12 (1984), pp. 1215-1230.
23. p-best monotone approximations, J. Statist. Plann. Inf. 47 (1995), pp. 295-318.
24. V. Chernozhukov, I. Fernandez-Val, and A. Galichon, Improving estimates of monotone functions by rearrangement. Preprint (2007), arXiv:0704.3686. Available at http://arxiv.org/abs/0704.3686v1.
25. N. Neumeyer, A note on uniform consistency of monotone function estimators, Statist. Probab. Lett. 77 (2007), pp. 693-703.
26. Y.P. Mack and B.W. Silverman, Weak and strong uniform consistency of kernel regression estimates, Z. Wahrsch. Verw. Gebiete 61 (1982), pp. 405-415.