On rates of convergence for bayesian density estimation

Scandinavian Journal of Statistics

Volumn 34, Issue 3, 2007, Pages 626-642

  Scricciolo, Catia ^a  

^a BOCCONI UNIVERSITY   (Italy)

Author keywords

Density estimation; Histogram; Polygon; Posterior distribution; Rate of convergence

Indexed keywords

EID: 34547991298     PISSN: 03036898     EISSN: 14679469     Source Type: Journal    
DOI: 10.1111/j.1467-9469.2006.00540.x     Document Type: Article

References (27)

1. Abou-Jaoude, S. (1976). Conditions nécessaires et suffisantes de convergence L1 en probabilité de l'histogramme pour une densité. Ann. Inst. H. Poincaré Sect. B (N. S.) 12, 213 231.
2. Andreev, A. Arjas, E. (1996). A note on histogram approximation in Bayesian density estimation. In Bayesian statistics 5 (eds J. M. Bernardo, J. O. Berger, A. P. Dawid A. F. M. Smith 487 490. Oxford University Press, New York.
3. Barron, A., Schervish, M. J. Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27, 536 561.
4. Beirlant, J., Berlinet, A., Biau, G. Vajda, I. (2002). Divergence-type errors of smooth Barron-type density estimators. Test 11, 191 217.
5. Choudhuri, N., Ghosal, S. Roy, A. (2005). Bayesian methods for function estimation. In Handbook of statistics, Vol. 25 : Bayesian thinking, modeling and computation (eds D. K. Dey C. R. Rao 373 414. Elsevier Science, North-Holland.
6. 1 view. John Wiley & Sons, New York.
7. Efromovich, S. (1999). Nonparametric curve estimation. Methods, theory, and applications. Springer-Verlag, New York.
8. Gasparini, M. (1996). Bayesian density estimation via Dirichlet density processes. J. Nonparametr. Statist. 6, 355 366.
9. Genovese, C. R. Wasserman, L. (2000). Rates of convergence for the Gaussian mixture sieve. Ann. Statist. 28, 1105 1127.
10. Ghosal, S. (2001). Convergence rates for density estimation with Bernstein polynomials. Ann. Statist. 29, 1264 1280.
11. Ghosal, S. van der Vaart, A. W. (2001). Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities. Ann. Statist. 29, 1233 1263.
12. Ghosal, S., Ghosh, J. K. van der Vaart, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28, 500 531.
13. 2-divergence. Statistics 32, 31 57.
14. Hartigan, J. A. (1996). Bayesian histograms. In Bayesian statistics 5 (eds J. M. Bernardo, J. O. Berger, A. P. Dawid A. F. M. Smith 211 222. Oxford University Press, New York.
15. Hjort, N. L. (1996). Bayesian approaches to non- and semiparametric density estimation. In Bayesian statistics 5 (eds J. M. Bernardo, J. O. Berger, A. P. Dawid A. F. M. Smith 223 253. Oxford University Press, New York.
16. Kruijer, W. van der Vaart, A. (2005). Posterior convergence rates for Dirichlet mixtures of beta densities. In Stochastics section: selected preprints, WS2005-9. Department of Mathematics, Faculty of Sciences, Vrije Universiteit Amsterdam. Available at: http://www.math.vu.nl/sto/publications-en. html/stochastics/2005/WS2005-9.pdf.
17. Parthasarathy, K. R. (1967). Probability measures on metric spaces. Academic Press, New York-London.
18. Petrone, S. (1999a). Random Bernstein polynomials. Scand. J. Statist. 26, 373 393.
19. Petrone, S. (1999b). Bayesian density estimation using Bernstein polynomials. Canad. J. Statist. 27, 105 126.
20. Petrone, S. Wasserman, L. (2002). Consistency of Bernstein polynomial posteriors. J. Roy. Statist. Soc. Ser. B 64, 79 100.
21. Royden, H. L. (1988). Real analysis, 3rd edn. Macmillan Publishing Company, New York.
22. Scott, D. W. (1979). On optimal and data-based histograms. Biometrika 66, 605 610.
23. Scott, D. W. (1992). Multivariate density estimation. Theory, practice, and visualization. John Wiley & Sons, New York.
24. Shen, X. Wasserman, L. (2001). Rates of convergence of posterior distributions. Ann. Statist. 29, 687 714.
25. Silverman, B. W. (1986). Density estimation for statistics and data analysis. Chapman & Hall, London.
26. Wong, W. H. Shen, X. (1995). Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. Ann. Statist. 23, 339 362.
27. Yang, Y. Barron, A. (1999). Information-theoretic determination of minimax rates of convergence. Ann. Statist. 27, 1564 1599.