Deconvolution for an atomic distribution

Electronic Journal of Statistics

Volumn 2, Issue , 2008, Pages 265-297

  Van Es, Bert ^a   Gugushvili, Shota ^b   Spreij, Peter ^a  

^a UNIVERSITY OF AMSTERDAM   (Netherlands)

^b EURANDOM   (Netherlands)

Author keywords

Asymptotic normality; Atomic distribution; De Convolution; Kernel density estimator

Indexed keywords

EID: 77049085715     PISSN: 19357524     EISSN: None     Source Type: Journal    
DOI: 10.1214/07-EJS121     Document Type: Article

References (50)

1. J. Aitchison. On the distribution of a positive random variable having a discrete probability mass at the origin. J. Amer. Statist. Assoc., 50:901-908, 1955.
2. J.O. Berger. Statistical Decision Theory and Bayesian Analysis. Springer, New York, 2nd edition, 1985.
3. C. Butucea and A. Tsybakov. Sharp optimality for density deconvolution with dominating bias, I. Theor. Probab. Appl., 52:111-128, 2007.
4. C. Butucea and A. Tsybakov. Sharp optimality for density deconvolution with dominating bias, II. Theor. Probab. Appl., 52:336-349, 2007.
5. P. Carr and D.B. Madan. Option valuation using the Fast Fourier Transform. J. Comput. Finance, 2:61-73, 1998.
6. R. Carrol and P. Hall. Optimal rates of convergence for deconvoluting a density. J. Am. Statist. Assoc., 83:1184-1186, 1988.
7. E. Cator. Deconvolution with arbitrary smooth kernels. Statist. Probab. Lett., 54:205-215, 2001.
8. K.L. Chung. A Course in Probability Theory. Academic Press, New York, 3rd edition, 2003.
9. A. Delaigle. An alternative view of the deconvolution problem. To appear in Stat. Sinica, 2007.
10. A. Delaigle and I. Gijbels. Bootstrap bandwidth selection in kernel density estimation from a contaminated sample. Ann. Inst. Statist. Math., 56:19-47, 2003.
11. A. Delaigle and I. Gijbels. Comparison of data-driven bandwidth selection procedures in deconvolution kernel density estimation. Comp. Statist. Data Anal., 45:249-267, 2004.
12. A. Delaigle and I. Gijbels. Frequent problems in calculating integrals and optimizing objective functions: a case study in density deconvolution. Stat. Comp., 17:349-355, 2007.
13. A. Delaigle and P. Hall. On optimal kernel choice for deconvolution. Stat. Probab. Lett., 76:1594-1602, 2006.
14. A. Delaigle, P. Hall and A. Meister. On deconvolution with repeated measurements. Ann. Statist., 36:665-685, 2008.
15. L. Devroye. A Course in Density Estimation. Birkhäuser, Boston, 1987.
16. L. Devroye. A note on consistent deconvolution in density estimation. Canad. J. Statist., 17:235-239, 1989.
17. L. Devroye and L. Györfi. Nonparametric Density Estimation: The View. John Wiley&Sons, New York, 1985.
18. P.J. Diggle and P. Hall. Fourier approach to nonparametric deconvolution of a density estimate. J. R. Statist. Soc. B, 55:523-531, 1993.
19. J. Fan. Asymptotic normality for deconvolution kernel density estimators. Sankhya Ser., 53:97-110, 1991.
20. J. Fan. Global behaviour of kernel deconvolution estimates. Statist. Sinica, 1:541-551, 1991.
21. J. Fan. On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist., 19:1257-1272, 1991.
22. J. Fan. Deconvolution for supersmooth distributions. Canad. J. Statist., 20:155-169, 1992.
23. J. Fan and Y. Liu. A note on asymptotic normality for deconvolution kernel density estimators. Sankhya Ser., 59:138-141, 1997.
24. J. Fan and Y.K. Truong. Nonparametric regression with errors in variables. Ann. Statist., 21:1900-1925, 1993.
25. A. Feuerverger and R.A. Mureika. The empirical characteristic function and its applications. Ann. Statist., 5:88-97, 1977.
26. S. Gugushvili. Decompounding under Gaussian noise. arXiv:0711-0719 [math.ST], 2007.
27. W. Härdle. Smoothing Techniques with Implementation in S. Springer, New York, 1991.
28. S.H. Hesse. Data-driven deconvolution. J. Nonparametr. Statist., 10:343-373, 1999.
29. W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc., 58:13-30, 1963.
30. H. Holzmann and L. Boysen. Integrated square error asymptotics for supersmooth deconvolution. Scand. J. Statist., 33:849-860, 2006.
31. B.G. Lindsay. Mixture Models: Theory, Geometry and Applications. IMS, Hayward, 1995.
32. M.C. Liu and R.L. Taylor. A consistent nonparametric density estimator for the deconvolution problem. Canad. J. Statist., 17:427-438, 1989.
33. J.S. Maritz and T. Lwin. Empirical Bayes Methods. Chapman and Hall, London, 2nd edition, 1989.
34. R.C. Merton. Option pricing when underlying stock returns are discontinuous. J. Financ. Econ., 3:125-144, 1976.
35. A. Meister. Density estimation with normal measurement error with un-known variance. Stat. Sinica 16:195-211, 2006.
36. B.L.S. Prakasa Rao. Nonparametric Functional Estimation. Academic Press, Orlando, 1983.
37. K.-I. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge, 2004.
38. L.A. Stefanski. Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett., 9:229-235, 1990.
39. L.A. Stefanski and R.J. Caroll. Deconvoluting kernel density estimators. Statistics, 21:169-184, 1990.
40. A. Tsybakov. Introduction à l’estimation non-paramétrique. Springer, Berlin, 2004.
41. A.W. van der Vaart. Asymptotic Statistics. Cambridge University Press, Cambridge, 1998.
42. B. van Es and S. Gugushvili. Some thoughts on the asymptotics of the deconvolution kernel density estimator. arXiv:0801.2600 [stat.ME], 2008.
43. B. van Es and S. Gugushvili. Weak convergence of the supremum distance for supersmooth kernel deconvolution. arXiv:0802.2186 [math.ST], 2008.
44. B. van Es, S. Gugushvili, and P. Spreij. A kernel type nonparametric density estimator for decompounding. Bernoulli, 13:672-694, 2007.
45. A.J. van Es and H.-W. Uh. Asymptotic normality of nonparametric kernel type estimators: crossing the Cauchy boundary. J. Nonparametr. Statist., 16:261-277, 2004.
46. A.J. van Es and H.-W. Uh. Asymptotic normality of kernel-type deconvolution estimators. Scand. J. Statist., 32:467-483, 2005.
47. M.P. Wand. Finite sample performance of deconvolving density estimators. Statist. Probab. Lett., 37:131-139, 1998.
48. M.P. Wand and M.C. Jones. Kernel Smoothing. Chapman & Hall, London,1995.
49. L. Wasserman. All of Nonparametric Statistics. Springer, Berlin, 2007.
50. C.H. Zhang. Fourier methods for estimating mixing densities and distributions. Ann. Statist., 18:806-831, 1990.