Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations

Annals of Statistics

Volumn 26, Issue 5, 1998, Pages 2014-2048

  Neumann, Michael H ^a  

^a HUMBOLDT UNIVERSITY   (Germany)

Author keywords

Bootstrap; Density estimation; Mixing; Nonparametric tests; Simultaneous confidence bands; Strong approximation; Weak dependence; Whitening by windowing

Indexed keywords

EID: 0032263989     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1024691367     Document Type: Article

References (45)

1. AMOSOVA, N. N. (1972). On limit theorems for probabilities of moderate deviations. Vestnik Leningrad. Univ. 13 5-14 (in Russian).
2. ANGO NZE, P. (1992). Critères d'ergodicité de quelques modèles à représentation markovienne. C. R. Acad. Sci. Paris Sèr. I Math. 315 1301-1304.
3. ANGO NZE, P. and DOUKHAN, P. (1993). Estimation fonctionnelle de séries temporelles mélangeantes. C. R. Acad. Sci. Paris Sèr. I Math. 317 405-408.
4. ∞ de la fonction de densité d'un processus faiblement dépendant: les cas absolument régulier et fortement mélangeant. C. R. Acad. Sci. Paris Sèr. I Math. 320 1259-1262.
5. BICKEL, P. and ROSENBLATT, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071-1095.
6. BRÉMAUD, P. and MASSOULIÉ, L. (1996). Stability of nonlinear Hawkes processes. Ann. Probab. 24 1563-1588.
7. BÜHLMANN, P. (1994). Blockwise bootstrapped empirical process for stationary sequences. Ann. Statist. 22 995-1012.
8. DHOMPONGSA, S. (1984). A note on the almost sure approximation of the empirical process of weakly dependent random vectors. Yokohama Math. J. 32 113-121.
9. DOUKHAN, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statist. 85. Springer, New York.
10. DOUKHAN, P., MASSART, P. and RIO, E. (1995). Invariance principles for absolutely regular empirical processes. Ann. Inst. H. Poincaré Probab. Statist. 31 393-427.
11. DOUKHAN, P. and PORTAL, F. (1987). Principe d'invariance faible pour la fonction de répartition empirique dans un cadre multidimensionnel et mélangeant. Probab. Math. Statist. 8 117-132.
12. EFRON, B. (1979). Bootstrap methods: another look at the jackknife. Ann. Statist. 7 1-26.
13. EFRON, B. (1982). The Jackknife, the Bootstrap and Other Resampling Plans. SIAM, Philadelphia.
14. FALK, M. and REISS, R.-D. (1989a). Weak convergence of smoothed and nonsmoothed boot-strap quantile estimates. Ann. Probab. 17 362-371.
15. FALK, M. and REISS, R.-D. (1989b). Bootstrapping the distance between smooth bootstrap and sample quantile distribution. Probab. Theory Related Fields 82 177-186.
16. FARAWAY, J. J. and JHUN, M (1990). Bootstrap choice of bandwidth for density estimation. J. Amer. Statist. Assoc. 85 1119-1122.
17. FRANKE, J., KREISS, J.-P., MAMMEN, E. and NEUMANN, M. H. (1998). Properties of the nonparametric autoregressive bootstrap. Discussion Paper 54/98, SFB 373, Humboldt Univ., Berlin.
18. FRIEDMAN, J., STUETZLE, W. and SCHROEDER, A. (1984). Projection pursuit density estimation. J. Amer. Statist. Assoc. 79 599-608.
19. GYÖRFI, L., HÄRDLE, W., SARDA, P. and VIEU, P. (1989). Nonparametric Curve Estimation from Time Series. Springer, Berlin.
20. HALL, P. (1991). On convergence rates of suprema. Probab. Theory Related Fields 89 447-455.
21. HALL, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
22. HALL, P. (1993). On Edgeworth expansion and bootstrap confidence bands in nonparametric curve estimation. J. R. Statist. Soc. Ser. B 55 291-304.
23. HALL, P., DICICCIO, T. J. and ROMANO, J. P. (1989). On smoothing and the bootstrap. Ann. Statist. 17 692-704.
24. HALL, P. and HART, J. D. (1990). Convergence rates in density estimation for data from infinite-order moving average processes. Probab. Theory Related Fields 87 253-274.
25. HÄRDLE, W. and MAMMEN, E. (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21 1926-1947.
26. HART, J. D. (1996). Some automated methods of smoothing time-dependent data. J. Non-parametr. Statist. 6 115-142.
27. KOMLÓS, J., MAJOR, P. and TUSNÁDY, G. (1975). An approximation of partial sums of independent rv's and the sample distribution function. Z. Wahrsch. Verw. Gebiete 32 111-131.
28. KONAKOV, V., LÄUTER, H. and LIERO, H. (1995). Nonparametric versus parametric goodness of fit. Statistics. To appear.
29. KÜNSCH, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217-1241.
30. MASRY, E. (1994). Probability density estimation from dependent observations using wavelet orthonormal bases. Statist. Probab. Lett. 21 181-194.
31. MOKKADEM, A. (1988). Mixing properties of ARMA processes. Stochastic Process. Appl. 29 309-315.
32. MOKKADEM, A. (1990). Propriétés de mélange des processus autorégessifs polynomiaux. Ann. Inst. H. Poincaré Probab Statist. 26 219-260.
33. NEUMANN, M. H. (1995). Automatic bandwidth choice and confidence intervals in nonparametric regression. Ann. Statist. 23 1937-1959.
34. NEUMANN, M. H. and KREISS, J.-P. (1998). Regression-type inference in nonparametric autoregression. Ann. Statist. 26 1570-1613.
35. NEUMANN, M. H. and PAPARODITIS, E. (1998). A nonparametric test for the stationary density. Discussion Paper 58/98, SFB 373, Humboldt Univ., Berlin.
36. PHAM, T. D. (1986). The mixing property of bilinear and generalised random coefficient autoregressive models. Stochastic Process. Appl. 23 291-300.
37. PHAM, T. D. and TRAN, L. T. (1985). Some mixing properties of time series models. Stochastic Process. Appl. 19 297-303.
38. REISS, R.-D. (1993). A Course on Point Processes. Springer, New York.
39. ROBINSON, P. M. (1983). Nonparametric estimators for time series. J. Time Ser. Anal. 4 185-207.
40. SHORACK, G. R. and WELLNER, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
41. SILVERMAN, B. W. and YOUNG, G. A. (1987). The bootstrap: to smooth or not to smooth? Biometrika 74 469-479.
42. SPOKOINY, V. G. (1996). Adaptive and spatially adaptive testing of a nonparametric hypothesis. Math. Methods Statist. To appear.
43. TAKAHATA, H. and YOSHIHARA, K. (1987). Central limit theorems for integrated square error of nonparametric density estimators based on absolutely regular random sequences. Yokohama Math. J. 35 95-111.
44. TRAN, L. T. (1990). Density estimation for time series by histograms. J. Statist. Plann. Inference 40 61-79.
45. ∞ norm for dependent data with applications to the Gibbs sampler. Ann. Statist. 21 711-735.