Multiscale maximum likelihood analysis of a semiparametric model, with applications

Annals of Statistics

Volumn 29, Issue 5, 2001, Pages 1297-1319

  Walther, Guenther ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

Adaptive; Failure rate; Grenander estimator; Log concave; Minimax; Mixture; Multiscale analysis; Penalized maximum likelihood

Indexed keywords

EID: 0035470892     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1013203455     Document Type: Article

References (36)

1. BROOKS, S. P. (1998). MCMC convergence diagnosis via multivariate bounds on log-concave densities. Ann. Statist. 26 398-433.
2. CHAUDHURI, P. and MARRON, J. S. (1999). SiZer for exploration of structures in curves. J. Amer. Statist. Assoc. 94 807-823.
3. CHERNOFF, H. (1964). Estimation of the mode. Ann. Inst. Statist. Math. 16 31-41.
4. CHOW, Y. S. and TEICHER, H. (1988). Probability Theory, 2nd ed. Springer, New York.
5. DELLAPORTAS, P. and SMITH, A. F. M. (1993). Bayesian inference for generalized linear and proportional hazards models via Gibbs sampling. J. Roy. Statist. Soc. Ser. C 42 443-460.
6. DÜMBGEN, L. and SPOKOINY, V. G. (2001). Multiscale testing of qualitative hypotheses. Ann. Statist. 29 124-152.
7. GILKS, W. R. and WILD, P. (1992). Adaptive rejection sampling for Gibbs sampling. J. Roy. Statist. Soc. Ser. C 41 337-348.
8. GLASER, R. E. (1980). Bathtub and related failure rate characteristics. J. Amer. Statist. Assoc. 75 667-672.
9. GREEN, P. J. and SILVERMAN, B. W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman and Hall, London.
10. 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, Belmont, CA.
11. GROENEBOOM, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probab. Theory related Fields 81 79-109.
12. GROENEBOOM, P., HOOGHIEMSTRA, G. and LOPUAÄ, H. P. (1999). Asymptotic normality of the L1 error of the Grenander estimator. Ann. Statist. 27 1316-1347.
13. GROENEBOOM, P., JONGBLOED, G. and WELLNER, J. A. (2001). Estimation of a convex function: characterizations and asymptotic theory. Ann. Statist. To appear.
14. HILDENBRAND, K. and HILDENBRAND W. (1985). On the mean income effect: a data analysis of the U.K. family expenditure survey. In Contributions to Mathematical Economics (W. Hildenbrand and A. Mas-Colell, eds.) 247-263. North-Holland, Amsterdam.
15. INGSTER, Y. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives I-III. Math. Methods Statist. 2 85-114, 171-189, 249-337.
16. JONGBLOED, G. (1998). The iterative convex minorant algorithm for nonparametric estimation. J. Comput. Graph. Statist. 7 310-321.
17. KIM, J. and POLLARD, D. (1990). Cube root asymptotics. Ann. Statist. 18 191-219.
18. KOMLÓS, J., MAJOR, P. and TUSNÁDY, G. (1975). An approximation of partial sums of independent r. v.'s and the sample d.f. Z. Wahrsch. Verw. Gebiete 32 111-131.
19. KOROSTELEV, A. and NUSSBAUM, M. (1995). Density estimation in the uniform norm and white noise approximation. Preprint 154, Weierstrass Institute, Berlin.
20. LIERO, H., LÄUTER, H. and KONAKOV, V. (1998). Nonparametric versus parametric goodness of fit. Statistics 31 115-149.
21. LINDSAY, B. G. and ROEDER, K. (1992). Residual diagnostics for mixture models. J. Amer. Statist. Assoc. 87 785-794.
22. LINDSAY, B. G. and ROEDER, K. (1997). Moment-based oscillation properties of mixture models. Ann. Statist. 25 378-386.
23. MILLER, R. G. (1981). Survival Analysis. Wiley, New York.
24. NEUMANN, M. H. (1998). Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations. Ann. Statist. 26 2014-2048.
25. PRAKASA RAO, B. L. S. (1969). Estimation of a unimodal density. Sankhyā Ser. A 31 23-36.
26. ROBERTSON, T., WRIGHT, F. T. and DYKSTRA, R. L. (1988). Order Restricted Statistical Inference. Wiley, New York.
27. ROEDER, K. (1994). A graphical technique for determining the number of components in a mixture of normals. J. Amer. Statist. Assoc. 89 487-495.
28. SHEATHER, S. J. and JONES, M. C. (1991). A reliable data-based bandwidth selection method for kernel density-estimation. J. Roy. Statist. Soc. Ser. B 53 683-690.
29. SHORACK, G. R. and WELLNER, J. A. (1986). Empirical Processes with applications to statistics. Wiley, New York.
30. SILVERMAN, B. W. (1982). On the estimation of a probability density function by the maximum penalized likelihood method. Ann. Statist 10 795-810.
31. SILVERMAN, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.
32. TITTERINGTON, D. M., SMITH, A. F. M. and MAKOV, U. E. (1985). Statistical Analysis of Finite Mixture Distributions. Wiley, New York.
33. WALTHER, G. (2000a). Multiscale maximum likelihood analysis of a semiparametric model, with applications. Technical report, Dept. Statistics, Stanford Univ.
34. WALTHER, G. (2000b). Detecting the presence of mixing with multiscale maximum likelihood. J. Amer. Statist. Assoc. To appear.
35. WOODROOFE, M. and SUN, J. (1999). Testing uniformity versus a monotone density. Ann. Statist. 27 338-360.
36. ZHANG, Y. L. and NEWTON, M. A. (1997). On calculating the nonparametric maximum likelihood estimator of a distribution given interval censored data. J. Comput. Graph. Statist. To appear.