Kernel density estimation via diffusion

Annals of Statistics

Volumn 38, Issue 5, 2010, Pages 2916-2957

  Botev, Z I ^a   Grotowski, J F ^a   Kroese, D P ^a  

^a UNIVERSITY OF QUEENSLAND   (Australia)

Author keywords

Bandwidth selection; Boundary bias; Data sharpening; Diffusion equation; Heat kernel; Langevin process; Nonparametric density estimation; Normal reference rules; Variable bandwidth

Indexed keywords

EID: 77956220369     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AOS799     Document Type: Article

References (53)

1. ABRAMSON, I. S. (1982). On bandwidth variation in kernel estimates-a square root law. Ann. Statist. 10 1217-1223. MR0673656
2. AZENCOTT, R. (1984). Density of diffusions in small time: Asymptotic expansions. In Seminar on Probability, XVIII. Lecture Notes in Math. 1059 402-498. Springer, Berlin. MR0770974
3. BELLMAN, R. (1961). A Brief Introduction to Theta Functions. Holt, Rinehart and Winston, New York. MR0125252
4. BOTEV, Z. I. (2007). Kernel density estimation using Matlab. Available at http://www. mathworks.us/matlabcentral/fileexchange/authors/27236.
5. BOTEV, Z. I. (2007). Nonparametric density estimation via diffusion mixing. Technical report, Dept. Mathematics, Univ. Queensland. Available at http://espace.library.uq.edu.au.
6. CHAUDHURI, P. and MARRON, J. S. (2000). Scale space view of of curve estimation. Ann. Statist. 28 408-428. MR1790003
7. CHOI, E. and HALL, P. (1999). Data sharpening as a prelude to density estimation. Biometrika 86 941-947. MR1741990
8. COHEN, J. K., HAGIN, F. G. and KELLER, J. B. (1972). Short time asymptotic expansions of solutions of parabolic equations. J. Math. Anal. Appl. 38 82-91. MR0303086
9. CSISZáR, I. (1972). A class of measures of informativity of observation channels. Period. Math. Hungar. 2 191-213. MR0335152
10. DEVRÔYE, L. (1997). Universal smoothing factor selection in density estimation: Theory and practice. Test 6 223-320. MR1616896
11. DOUCET, A., DE FREITAS, N. and GORDON, N. (2001). Sequential Monte Carlo Methods in Practice. Springer, New York. MR1847783
12. ETHIER, S. N. and KURTZ, T. G. (2009). Markov Processes. Characterization and Convergence. Wiley, New York. MR0838085
13. FELLER, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math. (2) 55 468-519. MR0047886
14. FRIEDMAN, A. (1964). Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs, NJ. MR0181836
15. HALL, P. (1990). On the bias of variable bandwidth curve estimators. Biometrika 77 523-535. MR1087843
16. HALL, P., HU, T. C. andMARRON, J. S. (1995). Improved variable window kernel estimates of probability densities. Ann. Ststist. 23 1-10. MR1331652
17. HALL, P. and MARRON, J. S. (1987). Estimation of integrated squared density derivatives. Statist. Probab. Lett. 6 109-115. MR0907270
18. HALL, P. and MINNOTTE, M. C. (2002). High order data sharpening for density estimation. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 141-157. MR1883130
19. HALL, P. and PARK, B. U. (2002). New methods for bias correction at endpoints and boundaries. Ann. Statist. 30 1460-1479. MR1936326
20. HALL, P. and PARK, B. U. (2002). New methods for bias correction at endpoints and boundaries. Ann. Statist. 30 1460-1479. MR1936326
21. HAVRDA, J. H. and CHARVAT, F. (1967). Quantification methods of classification processes: Concepts of structural a entropy. Kybernetika (Prague) 3 30-35. MR0209067
22. JONES, M. C. and FOSTER, P. J. (1996). A simple nonnegative boundary correction method for kernel density estimation. Statist. Sinica 6 1005-1013. MR1422417
23. JONES, M. C., MARRON, J. S. and PARK, B. U. (1991). A simple root n bandwidth selector. Ann. Statist. 19 1919-1932. MR1135156
24. JONES, M. C., MARRON, J. S. and SHEATHER, S. J. (1993). Simple boundary correction for kernel density estimation. Statist. Comput. 3 135-146.
25. JONES, M. C., MARRON, J. S. and SHEATHER, S. J. (1996). A brief survey of bandwidth selection for density estimation. J. Amer. Statist. Assoc. 91 401-407. MR1394097
26. JONES, M. C.,MARRON, J. S. and SHEATHER, S. J. (1996). Progress in data-based bandwidth selection for kernel density estimation. Comput. Statist. 11 337-381. MR1415761
27. JONES, M. C., MCKAY, I. J. and HU, T. C. (1994). Variable location and scale kernel density estimation. Ann. Inst. Statist. Math. 46 521-535. MR1309722
28. JONES, M. C. and SIGNORINI, D. F. (1997). A comparison of higher-order bias kernel density estimators. J. Amer. Statist. Assoc. 92 1063-1073. MR1482137
29. KANNAI, Y. (1977). Off diagonal short time asymptotics for fundamental solutions of diffusion equations. Comm. Partial Differential Equations 2 781-830. MR0603299
30. KAPUR, J. N. and KESAVAN, H. K. (1987). Generalized Maximum Entropy Principle (With Applications). Standford Educational Press, Waterloo, ON. MR0934205
31. KARUNAMUNI, R. J. and ALBERTS, T. (2005). A generalized reflection method of boundary correction in kernel density estimation. Canad. J. Statist. 33 497-509. MR2232376
32. KARUNAMUNI, R. J. and ZHANG, S. (2008). Some improvements on a boundary corrected kernel density estimator. Statist. Probab. Lett. 78 499-507. MR2400863
33. KERM, P. V. (2003). Adaptive kernel density estimation. Statist. J. 3 148-156.
34. KLOEDEN, P. E. and PLATEN, E. (1999). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.
35. LADYŽENSKAJA, O. A., SOLONNIKOV, V. A. and URAL'CEVA, N. N. (1967). Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23 xi+648. Amer. Math. Soc., Providence, RI. MR0241822
36. LARSSON, S. and THOMEE, V. (2003). Partial Differential Equations with Numerical Methods. Springer, Berlin. MR1995838
37. LEHMANN, E. L. (1990). Model specification: The views of fisher and neyman, and later developments. Statist. Sci. 5 160-168. MR1062574
38. LOADER, C. R. (1999). Bandwidth selection: Classical or plug-in. Ann. Statist. 27 415-438. MR1714723
39. LOFTSGAARDEN, D. O. and QUESENBERRY, C. P. (1965). A nonparametric estimate of a multivariate density function. Ann. Math. Statist. 36 1049-1051. MR0176567
40. MARRON, J. S. (1985). An asymptotically efficient solution to the bandwidth problem of kernel density estimation. Ann. Statist. 13 1011-1023. MR0803755
41. MARRON, J. S. and RUPPERT, D. (1996). Transformations to reduce boundary bias in kernel density-estimation. J. Roy. Statist. Soc. Ser. B 56 653-671. MR1293239
42. MARRON, J. S. andWAND, M. P. (1992). Exact mean integrated error. Ann. Statist. 20 712-736. MR1165589
43. MOLCHANOV, S. A. (1975). Diffusion process and Riemannian geometry. Russian Math. Surveys 30 1-63.
44. PARK, B. U., JEONG, S. O. and JONES, M. C. (2003). Adaptive variable location kernel density estimators with good performance at boundaries. J. Nonparametr. Stat. 15 61-75. MR1958960
45. PARK, B. U. and MARRON, J. S. (1990). Comparison of data-driven bandwidith selectors. J. Amer. Statist. Assoc. 85 66-72.
46. SAMIUDDIN, M. and EL-SAYYAD, G. M. (1990). On nonparametric kernel density estimates. Biometrika 77 865. MR1086696
47. SCOTT, D. W. (1992). Multivariate Density Estimation. Theory, Practice and Visualization. Wiley, New York. MR1191168
48. 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. MR1125725
49. SILVERMAN, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London. MR0848134
50. SIMONOFF, J. S. (1996). Smoothing Methods in Statistics. Springer, New York. MR1391963
51. TERRELL, G. R. and SCOTT, D. W. (1992). Variable kernel density estimation. Ann. Statist. 20 1236-1265. MR1186249
52. WAND, M. P. and JONES, M. C. (1994). Multivariate plug-in bandwidth selection. Comput. Statist. 9 97-117. MR1280754
53. WAND, M. P. and JONES, M. C. (1995). Kernel Smoothing. Chapman and Hall, London. MR1319818