Multivariate mode hunting: Data analytic tools with measures of significance

Journal of Multivariate Analysis

Volumn 100, Issue 6, 2009, Pages 1198-1218

  Burman, Prabir ^a   Polonik, Wolfgang ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

62G20; 62G99; 62H99; Brownian bridge; Distribution free; Modality; Nearest neighbor method; primary; secondary; Testing for antimodes; VC classes

Indexed keywords

EID: 62349086715     PISSN: 0047259X     EISSN: 10957243     Source Type: Journal    
DOI: 10.1016/j.jmva.2008.10.015     Document Type: Article

References (41)

1. Corti S., Molteni F., and Palmer P.N. Signature of recent climate change in frequencies of natural atmospheric circulation regimes. Nature 398 (1999) 799-802
2. Hartigan J.A. Clustering Algorithms (1975), Wiley, New York
3. Jiang W. Nonparametric density estimation and clustering in astronomical sky surveys. Comput. Statist. Data Anal. 50 (2006) 760-774
4. Cuevas A., Febrero M., and Fraiman R. Estimating the numbers of clusters. Canad. J. Statist. 28 (2000) 367-382
5. Friedman J., and Fisher N.I. Bump hunting in high-dimensional data (with discussion). Comput. Statist. 9 (1999) 123-143
6. W. Polonik, Z. Wang, PRIM Anal. (2006) (submitted for publication)
7. Haldane J.B.S. Simple tests for bimodality and bitangentiality. Ann. Eugenics 14 (1952) 309-318
8. Dalenius T. The mode: a neglected statistical parameter. J. Roy. Statist. Soc. Ser. A 128 (1965) 110-117
9. Venter J.H. On estimation of the mode. Ann. Math. Statist. 38 (1967) 1446-1455
10. Lientz B.P. Results on nonparametric modal intervals. SIAM J. Appl. Math. 19 (1970) 356-366
11. Good I.J., and Gaskins R.A. Density estimation and bump-hunting by the penalized maximum likelihood method exemplified by scattering and meteorite data. (with discussion). J. Amer. Statist. Assoc. 75 (1980) 42-73
12. Silverman B. Using kernel density estimates to investigate multimodality. J. Roy. Statist. Soc. Ser. B 43 (1981) 97-99
13. Hartigan J.A., and Hartigan P.M. The dip test of unimodality. Ann. Statist. 13 (1985) 70-84
14. Wong M.A. A bootstrap testing procedure for investigating the number of subpopulations. J. Statist. Comput. Simul. 22 (1985) 99-112
15. M.A. Wong, C. Schaack, (1985): Diagnostics for assessing multimodality. in: L. Billard (Ed.), Proceedings of the Sixteenth Symposium on the Interface, in: Computer science and statistics, pp. 287-296
16. Comparini A., and Gori E. Estimating modes and antimodes of multimodal densities. Metron 44 (1986) 307-332
17. Müller H.-G. Adaptive nonparametric peak estimation. Ann. Statist. 17 (1989) 1053-1069
18. Müller D.W., and Sawitzki G. Excess mass estimates and tests for multimodality. J. Amer. Statist. Assoc. 86 (1991) 738-746
19. Roeder K. A graphical technique for determining the number of components in a mixture of normals. J. Amer. Statist. Assoc. 89 (1994) 487-495
20. Cheng M.-Y., and Hall P. Mode testing in difficult cases. Ann. Statist. 27 (1999) 1294-1315
21. Hall P., and Cheng M.Y. Mode testing in difficult cases. Ann. Statist. 27 (1999) 1294-1315
22. Bai Z.D., and Huang L.D. On consistency of the best-r-points-average estimator for the location for the maximizer of a nonparametric regression function. Sankhya: Indian J. Statist, Ser. A 61 (1999) 208-217
23. Walther G. Multiscale maximum likelihood analysis of a semiparametric model, with applications. Ann. Statist. 29 (2001) 1297-1319
24. Bai Z., and Chen Z. Convergence rate of the best-r-point-average estimator for the maximizer of a nonparametric regression function. J. Multivariate Anal. 84 (2003) 319-334
25. Hartigan J.A. The span test for unimodality. In: Bock H.H. (Ed). Classification and related methods of data analysis (1988), North-Holland, Amsterdam 229-236
26. Hartigan J.A., and Mohatny S. The RUNT test for multimodality. J. Classification 9 (1992) 115-132
27. Rozal P.M., and Hartigan J.A. The MAP test for multimodality. J. Classification 11 (1994) 5-36
28. Minotte M.C. Nonparametric testing of the existence of modes. Ann. Statist. 25 (1997) 1646-1660
29. Polonik W. Measuring mass concentration and estimating density contour clusters: An excess mass approach. Ann. Statist. 23 (1995) 855-881
30. Polonik W. Minimum volume sets and generalized quantile processes. Stochastic Process. Appl. 69 (1997) 1-24
31. Tsybakov A.B. On nonparametric estimation of level sets. Ann. Statist. 25 (1997) 948-969
32. Walther G. Granulometric smoothing. Ann. Statist. 25 (1997) 2273-2299
33. Molchanov I. A limit theorem for solutions of inequalities. Scand. J. Statist. 25 (1998) 235-242
34. A. Cuevas, W. Gonzáles-Manteiga, A. Rodriguez-Casal, Plug in estimation of general level sets, 2004. Manuscript
35. Polonik W., and Wang Z. Estimation of regression contour clusters: An application of the excess mass approach to regression. J. Multivariate Anal. 94 (2005) 227-249
36. Burman P., and Nolan D. Location-adaptive density estimation and nearest neighbor distance. J. Multivariate Anal. 40 (1992) 132-157
37. Alexander K.S. Probability inequalities for empirical processes and a law of the iterated logarithm. Ann. Probab. 12 (1984) 1041-1067 (Correction: 15 428-430)
38. van de Geer S. Empirical Processes in M-estimation (2000), Cambridge University Press
39. van der Vaart A.W., and Wellner J.A. Weak Convergence and Empirical Processes (1996), Springer, New York
40. Simonoff J.S. Smoothing Methods in Statistics (1996), Springer, New York
41. Gayraud G., and Rousseau J. Rates of convergence for Bayesian level set estimation. Scand. J. Statist. 32 (2005) 639-660