Berry-esseen bounds for smooth estimator of a distribution function under association

Journal of Nonparametric Statistics

Volumn 11, Issue 1-3, 1999, Pages 79-106

  Cai, Zongwu ^a   Roussas, George G ^b  

^a UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Asymptotic normality; Berry Esseen inequality; Negative association; Positive association; Smooth estimate; Stationarity

Indexed keywords

EID: 0005979233     PISSN: 10485252     EISSN: None     Source Type: Journal    
DOI: 10.1080/10485259908832776     Document Type: Article

References (38)

1. Azzalini, A. (1981) A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika, 68, 326-328.
2. Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, New York.
3. Birkel, T. (1988a) Moment bounds for associated sequences. Ann. Probab., 16, 1184-1193.
4. Birkel, T. (1988b) On the convergence rate in the central limit theorem for associated processes. Ann. Probab., 16, 1685-1698.
5. Brindley, E. C. Jr. and Thompson, W. A. Jr. (1972) Dependence and aging aspects of multivariate survival. J. Amer. Statist. Assoc., 67, 822-830.
6. Cai, Z. (1993) Uniform strong convergence and rates for the kernel estimators of a distribution function and a regression function under weakly dependent assumptions. Chinese J. Applied Probab. Statist., 9, 11-17.
7. Cai, Z. and Roussas, G. G. (1992) Uniform strong estimation under α-mixing, with rates. Statist. Probab. Lett., 15, 47-55.
8. Cai, Z. and Roussas, G. G. (1995) Berry-Esseen bounds for smooth estimator of a distribution function under association. Technical Report No. 306, Division of Statistics, University of California-Davis.
9. Cai, Z. and Roussas, G. G. (1997a) Efficient estimation of a distribution function under quadrant dependence. Scand. J. Statist., 25, 211-224.
10. Cai, Z. and Roussas, G. G. (1997b) Smooth estimator of quantiles under association. To appear in: Statist. Probab. Letters.
11. Esary, J. D., Proschan, F. and Walkup, D. W. (1967) Association of random variables, with applications. Ann. Math. Statist., 38, 1466-1474.
12. Falk, M. (1983) Relative efficiency and deficiency of kernel type estimators of smooth distribution functions. Statist. Neerlandica, 37, 73-83.
13. Fortuin, C. M., Kastelyn, P. W. and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Commun. Math. Phys., 22, 89-103.
14. Hall, P. (1982) Rates of Convergence in the Central Limit Theorem, Pitman, London.
15. Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process. Proc. Camb. Phil. Soc., 56, 13-20.
16. Joag-Dev, K. (1983) Independence via uncorrelatedness under certain dependence structures. Ann. Probab., 11, 1037-1041.
17. Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Statist., 11, 286-295.
18. Lehmann, E. (1966) Some concepts of dependence. Ann. Math. Statist., 37, 1137-1153.
19. Lejeune, M. and Sarda, P. (1992) Smooth estimators of distribution and density functions. Comput. Statist. Data Anal., 14, 457-471.
20. Nadaraya, E. A. (1964) Some new estimates for distribution function. Theory Probab. Appl., 9, 497-500.
21. Newman, C. M. (1980) Normal fluctuations and the FKG inequalities. Commun. Math. Phys., 74, 119-128.
22. Newman, C. M. (1984) Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y. L. (Ed.), Inequality in Statistics and Probability, IMS Lecture Notes-Monograph Series, 5, 127-140.
23. Newman, C. M. (1990) Ising models and dependent percolation. In: Block, H. W., Sampson, A. R. and Savits, T. H. (Eds.), Topics in Statistical Dependence, IMS Lecture Notes-Monograph Series, 16, 395-401.
24. Reiss, R. D. (1981) Nonparametric estimation of smooth distribution functions. Scand. J. Statist., 8, 116-119.
25. Roussas, G. G. (1969) Nonparametric estimation of the transition distribution function of a Markov process. Ann. Math. Statist., 40, 1386-1400.
26. Roussas, G. G. (1993) Smooth estimation of the distribution function under association: Asymptotic normality. Technical Report No. 289, Division of Statistics, University of California, Davis.
27. Roussas, G. G. (1995) Asymptotic normality of a smooth estimate of a random field distribution function under association. Statist. Probab. Lett., 24, 77-90.
28. Sadikova, S. M. (1966) Two dimensional analogues of an inequality of Esseen with applications to the central limit theorem. Theory Probab. Appl., 11, 325-335.
29. Sarda, P. (1993) Smoothing parameter selection for smooth distribution functions. J. Statist. Plann. Inference, 35, 65-75.
30. Shaked, M. (1982) A general theory of some positive dependence notions. J. Multivariate Anal., 12, 199-218.
31. Shao, Q. M. (1986) A Berry-Esseen inequality and an invariance principle for associated random fields. Chinese J. Applied Probab. Statist., 5, 1-8.
32. Stout, W. F. (1974) Almost Sure Convergence, Academic Press, New York.
33. Tikhomirov, A. N. (1980) On the convergence rate in the central limit theorem for weakly dependent random variables. Theory Probab. Appl., 25, 790-809.
34. Watson, G. S. and Leadbetter, M. R. (1964) Hazard analysis II. Sankhyã Ser. A, 26, 101-116.
35. Winter, B. B. (1973) Strong uniform consistency of integrals of density estimators. Canad. J. Statist., 1, 247-253.
36. Winter, B. B. (1979) Convergence rate of perturbed empirical distribution functions. J. Appl. Probab., 16, 163-173.
37. Wood, T. (1983) A Berry-Esseen theorem for associated random variables. Ann. Probab., 4, 1042-1047.
38. Yamato, H. (1973) Uniform convergence of an estimator of a distribution function. Bull. Math. Statist., 15, 69-78.