Multivariate probability density estimation by wavelet methods: Strong consistency and rates for stationary time series

Stochastic Processes and their Applications

Volumn 67, Issue 2, 1997, Pages 177-193

  Masry, Elias ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Besov spaces; Probability density estimation; Rates of strong convergence; Strongly mixing processes; Wavelet method

Indexed keywords

EID: 0031574858     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(96)00005-1     Document Type: Article

References (42)

1. P. Ango Nze and P. Doukhan, Density estimation for mixing time series, C.R. Acad. Sci. Ser. I-Math 317 (1993) 405-408.
2. ∞ norm for weakly dependent processes: the absolutely regular and strongly mixing cases, C.R. Acad. Sci. Ser. I-Math. 320 (1995) 1259-1262.
3. L. Benova, D. Kendall and I. Stefanov, Spline transformation, J.R. Statistic. Soc. Ser. B 33 (1971) 1-70.
4. J. Bergh and J. Lofstrom, Interpolation Spaces - An Introduction Sprinter, New York, (1976).
5. R.C. Bradley, Approximation theorems for strongly mixing random variables, Michigan Math. J. 30 (1983) 69-81.
6. N.N. Cencov, Evaluation of an unknown distribution density from observations, Sov. Math. 3 (1962) 1559-1563.
7. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992).
8. D.L. Donoho, I.M. Johnstone, G. Kerkyacharian and D. Picard, Density estimation by wavelet thresholding, Ann. Statist. 24 (1996) 508-539.
9. P. Doukhan and J.R. Léon, Déviation quadratique d'estimateurs de densité par projections orthogonales, C.R. Acad. Sci. Paris 310, Ser. 1 (1990) 425-430.
10. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic Press, New York, 1972).
11. K. Fukunaga and L.D. Hostetler, The estimation of the gradient of a density function, with applications in pattern recognition, IEEE Trans. Inform. Theory 21 (1975) 32-40.
12. 2 strong consistency of recursive kernel density estimation from dependent samples, IEEE Trans. Inform Theory 36 (1990) 531-539.
13. P. Hall and C.C. Heyde, Martingale Limit Theory and Its Application (Academic Press, New York, 1980).
14. P. Hall and P. Patil, Formulae for mean integrated squares error of nonlinear wavelet-based density, Ann. Statist. 23 (1995) 905-928.
15. D. Ionnides and G. Roussas, Note on uniform convergence for mixing random variables, Statist. Probab. Lett. 5 (1987) 279-285.
16. G. Kerkyacharian and D. Picard, Density estimation is Besov spaces, Statist. Probab Lett. 13 (1992) 15-24.
17. M. Kon and L.A. Raphael, Characterizing convergence rates for multiresolution approximations, in: J. Zeevi and R. Coifman, eds., Signal and Image Representation in Combined Spaces (Academic Press, New York, 1993).
18. R. Kronmal and M. Tarter, The estimation of probability densities and cumulatives by Fourier series methods, J. Amer. Statist. Assoc. 63 (1968) 925-952.
19. A. Krzyzak and M. Pawlak, Distribution-free consistency of nonparametric kernel regression estimates and classification, IEEE Trans. Inform. Theory 30 (1984) 78-81.
20. A. Krzyzak, The rates of convergence of kernel regression estimates and classification rules, IEEE Trans. Inform. Theory 32 (1986) 668-679.
21. p loss, Statist. Probab. Lett. 27 (1996) 71-84.
22. K.S. Lii and M. Rosenblatt, Asymptotic behavior of a spline estimate of a density function, Comput. Math. Appl. 1 (1975) 232-235.
23. E. Masry, Asymptotic normality for deconvolution estimators of multivariate densities of stationary processes, J. Multivariate Anal. 44 (1993a) 47-68.
24. E. Masry, Strong consistency and rates for deconvolution of multivariate densities of stationary processes Stochastic Process. Appl. 47 (1993b) 53-74.
25. E. Masry, Probability density estimation from dependent observations using wavelet orthonormal bases, Statist. Probab. Lett. 21 (1994) 181-194.
26. Y. Meyer, Wavelets and Operators (Cambridge Univ. Press, Oxford, 1992).
27. E. Parzen, On the estimation of probability density and mode, Ann. Math. Statist. 33 (1962) 1065-1076.
28. P. Robinson, Nonparametric estimators for time series, J. Time Ser. Anal. 4 (1983) 185-297.
29. M. Rosenblatt, Remarks on some nonparametric estimates of density function, Ann. Math, Statist. 27 (1956a) 832-837.
30. M. Rosenblatt, A central limit theorem and strong mixing conditions, Proc. Nat. Acad. Sci. 4 (1956b) 43-47.
31. G. Roussas, Nonparametric estimation in mixing sequences of random variables, J. Statist. Planning Inference 18 (1988) 135-149.
32. D.W. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization (Wiley, New York, 1992).
33. B.W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman & Hall, New York, 1986).
34. S.C. Schwartz, Estimation of probability density by an orthogonal series, Ann. Math. Statist. 38 (1967) 1261-1265.
35. C.J. Stone, Optimal global rates of convergence of nonparametric regression, Ann. Statist. 10 (1982) 1040-1053.
36. L. Tran, Density estimation under dependence, Statist. Probab. Lett. 10 (1990) 193-201.
37. L. Tran, Kernel density estimation for linear processes, Stochastic Process. Appl. 41 (1992) 281-296.
38. H. Triebel, Theory of Function Space (Burkhäuser, Basel, 1983).
39. G. Wahba, Optimal convergence properties of variable knot, kernel and orthogonal series methods for density estimation, Ann. Statist. 3 (1975) 15-29.
40. G.G. Walter, Properties of Hermite series estimation of probability density, Ann. Statist. 5 (1977) 1258-1264.
41. G.G. Walter, Approximation of the delta function by wavelets, J. Approx. Theory 3 (1992) 329-343.
42. ∞-norm for dependent data with applications to the Gibbs sampler, Ann. Statist. 21 (1993) 711-735