Least squares estimators of the mode of a unimodal regression function

Annals of Statistics

Volumn 29, Issue 3, 2001, Pages 648-665

  Shoung, Jyh Ming ^a   Zhang, Cun Hui ^a  

^a RUTGERS UNIVERSITY   (United States)

Author keywords

Convergence rate; Least squares estimation; Mode; Nonparametric regression

Indexed keywords

EID: 0035539845     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1009210684     Document Type: Article

References (27)

1. AVER, 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. BICKEL, P. J. and FAN, J. (1996). Some problems on the estimation of unimodal densities. Statist. Sinica 3 23-45.
4. BIRGÉ, L. (1989). The Grenander estimator: A nonasymptotic approach. Ann. Statist. 17 1532-1549.
5. BIRGÉ, L. (1997). Estimation of unimodal densities without smoothness assumptions. Ann. Statist. 25 970-981.
6. BROWN, L. D. and LOW, M. G. (1996). A constrained risk inequality with applications to nonparametric functional estimation. Ann. Statist. 24 2524-2535.
7. BRUNK, H. D. (1970). Estimation of isotonic regression (with discussion by Ronald Pyke). In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 177-197. Cambridge Univ. Press.
8. CHERNOFF, H. (1964). Estimation of the mode. Ann. Inst. Stat. Math. 16 31-41.
9. DONOHO, D. L. and LIU, R. C. (1991). Geometrizing rates of convergence, II. Ann. Statist. 19 633-667.
10. EDDY, W. F. (1980). Optimum kernel estimators of the mode. Ann. Statist. 8 870-882.
11. GILL, R. D. and LEVIT, B. Y. (1995). Applications of the van Trees inequality: a Bayesian Cramér-Rao bound. Bernoulli 1 59-79.
12. GRENANDER, U. (1956). On the theory of mortality measurement, Part II. Skand. Akt. 39 125-153.
13. GROENEBOOM, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (L. M. Le Cam and R. A. Olshen, eds.) 2 539-555. Wadsworth, Belmont, CA.
14. p error in mode estimation. Ann. Statist. 23 2264-2284.
15. HASMINSKII, R. Z. (1979). Lower bound for the risks of nonparametric estimates of the mode. In Contribution to Statistics: Hajek Memorial Volume (J. Jureckova, ed.) 91-97. Academia, Prague.
16. LEDOUX, M. (1996). On Talagrand's deviation inequalities for product measures. ESAIM: Probability and Statistics 1 63-87.
17. LEPSKII, O. V. (1992). On problems of adaptive estimation in white Gaussian noise. In Advances in Soviet Mathematics 12. Amer. Math. Soc., Providence, RI.
18. MÜLLER, H.-G. (1989). Adaptive nonparametric peak estimation. Ann. Statist. 17 1053-1069.
19. PARZEN, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065-1076.
20. POLLARD, D. (1984). Convergence of Stochastic Processes. Springer, New York.
21. PHAKASA RAO, B. L. S. (1969). Estimation of a unimodal density. Sankhyā Ser. A 31 23-36.
22. ROBERTSON, T., WRIGHT, F. T. and DYKSTRA, R. L. (1988). Order Restricted Statistical Inference. Wiley, New York.
23. STONE, C. J. (1980). Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8 1348-1360.
24. VAN EEDEN, C. (1956). Maximum likelihood estimation of ordered probabilities. Proc. K. ned. Akad. Wet A 59 /Indag. Math. 18 444-455.
25. VAN EEDEN, C. (1956). Maximum likelihood estimation of ordered probabilities. Proc. K. ned. Akad. Wet A 59 /Indag. Math. 18 444-455.
26. VENTER, J. H. (1967). On estimation of the mode. Ann. Math. Statist. 38 1446-1455.
27. WEGMAN, E. J. (1970). Maximum likelihood estimation of a unimodal density function. Ann. Math. Statist. 41 457-471.