Estimating a convex function in nonparametric regression

Scandinavian Journal of Statistics

Volumn 34, Issue 2, 2007, Pages 384-404

  Birke, Melanie ^a   Dette, Holger ^a  

^a RUHR UNIVERSITY BOCHUM   (Germany)

Author keywords

Convexity; Local linear estimate; Nadaraya Watson estimate; Nondecreasing rearrangement; Nonparametric regression; Order restricted inference

Indexed keywords

EID: 34547317836     PISSN: 03036898     EISSN: 14679469     Source Type: Journal    
DOI: 10.1111/j.1467-9469.2006.00534.x     Document Type: Article

References (32)

1. Aït-Sahalia, Y. Duarte, J. (2003). Nonparametric option pricing under shape restrictions. J. Econometr. 116, 9 47.
2. Barlow, R. E., Bartholomew, D. J., Bremner, J. M. Brunk, H. D. (1972). Statistical inference under order restrictions: the theory and application of isotonic regression. Wiley, New York.
3. Bennett, C. Sharpley, R. (1988). Interpolation of operators. Academic Press, New York.
4. Birke, M. Dette, H. (2005). A note on estimating a monotone regression function by combining kernel and density estimates. Technical Report, Department of Mathematics. Available at: http://www.ruhr-uni-bochum.de/mathematik3/ preprint.htm.
5. Birke, M. Dette, H. (2006). Estimating a convex function in nonparametric regression. Technical Report, Department of Mathematics. Available at: http://www.ruhr-uni-bochum.de/mathematik3/preprint.htm.
6. Black, F. Scholes, M. (1973). The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637 654.
7. Brunk, H. D. (1955). Maximum likelihood estimates of monotone parameters. Ann. Math. Statist. 26, 607 616.
8. Cox, J. C., Ingersoll, J. E. Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385 407.
9. Dette, H., Neumeyer, N. Pilz, K. F. (2006). A simple nonparametric estimator of a monotone regression function. Bernoulli 12, 469 490.
10. Dudzinski, M. L. Mykytowycz, R. (1961). The eye lens as an indicator of age in the wild rabbit in Australia. CSIRO Wildl. Res. 6, 156 159.
11. Dykstra, R. L. (1983). An algorithm for restricted least squares regression. J. Amer. Statist. Assoc. 78, 837 842.
12. Fan, J. Gijbels, I. (1996). Local polynomial modelling and its applications. Chapman and Hall, London.
13. Fraser, D. A. S. Massam, H. (1989). A mixed primal-dual bases algorithm for regression under inequality constraints. Application to concave regression. Scand. J. Statist. 16, 65 74.
14. Gasser, T., Müller, H.-G. Mammitzsch, V. (1985). Kernels for nonparametric curve estimation. J. Roy. Statist. Soc. Ser. B 47, 238 252.
15. Groeneboom, P., Jongbloed, G. Wellner, J. A. (2001). Estimation of convex functions: characterizations and asymptotic theory. Ann. Statist. 26, 1653 1698.
16. Härdle, W. (1991). Applied nonparametric regression. Cambridge University Press, Cambridge.
17. Han, S. P. (1988). A successive projection method. Math. Program. 40, 1 14.
18. Hanson, P. L. Pledger, G. (1976). Consistency in concave regression. Ann. Statist. 4, 1038 1050.
19. Hildreth, C. (1954). Point estimates of ordinates of concave functions. J. Amer. Statist. Assoc. 49, 598 619.
20. Mack, Y. P. Silverman, B. W. (1982). Weak and strong uniform consistency of kernel regression estimates. Z. Wahrsch. Verw. Gebiete 61, 405 415.
21. Mammen, E. (1991). Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19, 741 759.
22. Müller, H.-G. (1985). Kernel estimators of zeros and of location and size of extrema of regression functions. Scand. J. Statist. 12, 221 232.
23. Mukerjee, H. (1988). Monotone nonparametric regression. Ann. Statist. 16, 741 750.
24. Nadaraya, E. A. (1964). On estimating regression. Theory Probab. Appl. 9, 141 142.
25. Neumeyer, N. (2006). A note on uniform consistency of monotone function estimators. Statist. Probab. Lett. (in press).
26. Ratkowsky, D. A. (1983). Nonlinear regression modeling. Marcel Dekker Inc., New York.
27. Rice, J. (1984). Bandwidth choice for nonparametric regression. Ann. Statist. 12, 1215 1230.
28. Robertson, T., Wright, F. T. Dykstra, R. L. (1988). Order restricted statistical inference. Wiley Series in Probability and Mathematical Statistics, Chichester.
29. Ryff, J. V. (1965). Orbits of L1 under doubly stochastic transformations. Trans. Am. Math. Soc. 117, 92 100.
30. Ryff, J. V. (1970). Measure preserving transformations and rearrangements. J. Math. Anal. Appl. 31, 449 458.
31. Watson, G. S. (1964). Smooth regression analysis. Sankhyā, Ser. A 26, 359 372.
32. Wu, C. F. (1982). Some algorithms for concave and isotonic regression. TIMS Stud. Manage. Sci. 19, 105 116.