Risk bounds in isotonic regression

Annals of Statistics

Volumn 30, Issue 2, 2002, Pages 528-555

  Zhang, Cun Hui ^a  

^a RUTGERS UNIVERSITY   (United States)

Author keywords

Isotonic regression; Least squares estimator; Maximum likelihood estimator; Nonparametric regression; Risk bounds

Indexed keywords

EID: 0036278582     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1021379864     Document Type: Article

References (29)

1. AYER, M., BRUNK, H. D., EWING, G. M., REID, W. T. and SILVERMAN, E. (1955). An empirical distribution function for sampling with incomplete information. Ann. Math. Statist. 26 641-647.
2. BARLOW, R. E., BARTHOLOMEW, D. J., BREMNER, J. M. and BRUNK, H. D. (1972). Statistical Inference Under Order Restrictions. Wiley, New York.
3. BARLOW, R. E. and UBHAYA, V. A. (1971). Isotonic approximation. In Optimizing Methods in Statistics (J. S. Rustagi, ed.) 77-86. Academic Press, New York.
4. BIRGÉ, L. (1987). Estimating a density under order restrictions: Nonasymptotic minimax risk. Ann. Statist. 15 995-1012.
5. BIRGÉ, L. (1989). The Grenander estimator: A nonasymptotic approach. Ann. Statist. 17 1532-1549.
6. BIRGÉ, L. and MASSART, P. (1993). Rates of convergence for minimum contrast estimators. Probab. Theory Related Fields 97 113-150.
7. BRUNK, H. D. (1970). Estimation of isotonic regression (with discussion by R. Pyke). In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 177-197. Cambridge Univ. Press, London.
8. BRUNK, H. D. and JOHANSEN, S. (1970). A generalized Radon-Nikodym derivative. Pacific J. Math. 34 585-617.
9. CHOW, Y. S. and LAI, T. L. (1975). Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossings. Trans. Amer. Math. Soc. 208 51-72.
10. CHOW, Y. S. and LAI, T. L. (1978). Paley-type inequalities and convergence rates related to the law of large numbers and extended renewal theory. Z. Wahrsch. Verw. Gebiete 45 1-19.
11. DONOHO, D. (1991). Gel'fand n-widths and the method of least squares. Technical report, Dept. Statist., Univ. California, Berkeley.
12. GRENANDER, U. (1956). On the theory of mortality measurement II. Skand. Aktuarietidskr. 39 125-153.
13. GRENANDER, U. (1980). Abstract Inference. Wiley, New York.
14. GROENEBOOM, P. (1983). The concave majorant of Brownian motion. Ann. Probab. 11 1016-1027.
15. GROENEBOOM, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (L. Le Cam and R. A. Olshen, eds.) 2 539-555. Wadsworth, New York.
16. GROENEBOOM, P. and WELLNER, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, New York.
17. HOFFMANN-JORGENSEN, J. (1974). Sums of independent Banach space valued random variables. Studio Math. 52 159-189.
18. MEYER, M. and WOODROOFE, M. (2000). On the degrees of freedom in shape-restricted regression. Ann. Statist. 28 1083-1104.
19. PRAKASA RAO, B. L. S. (1969). Estimation of a unimodal density. Sankhyā Ser. A 31 23-36.
20. ROBERTSON, T. and WALTMAN, P. (1968). On estimating monotone parameters. Ann. Math. Statist. 39 1030-1039.
21. ROBERTSON, T., WRIGHT, E. T. and DYKSTRA, R. L. (1988). Order Restricted Statistical Inference. Wiley, New York.
22. SERFLING, R. J. (1970). Moment inequalities for the maximum cumulative sum. Ann. Math. Statist. 41 1227-1234.
23. STEIN, C. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135-1151.
24. VAN DE GEER, S. (1990). Estimating a regression function. Ann. Statist. 18 907-924.
25. VAN DE GEER, S. (1993). Hellinger consistency of certain nonparametric maximum likelihood estimators. Ann. Statist. 2114-44.
26. VAN EEDEN, C. (1956). Maximum likelihood estimation of ordered probabilities. Indag. Math. 18 444-455.
27. VAN EEDEN, C. (1957a). Maximum likelihood estimation of partially or completely ordered parameters I. Indag. Math. 19 128-136.
28. VAN EEDEN, C. (1957b). Maximum likelihood estimation of partially or completely ordered parameters II. Indag. Math. 19 201-211.
29. 2 risk of an isotonic estimate. Comm. Statist. Theory Methods 25 281-294.