On the Uniform Strong Consistency of Local Polynomial Regression under Dependence Conditions

Communications in Statistics - Theory and Methods

Volumn 32, Issue 12, 2003, Pages 2415-2440

  Francisco Fernandez M ^a   Vilar Fernandez J M ^a   Vilar Fernandez J A ^a  

^a UNIVERSITY OF A CORUÑA   (Spain)

Author keywords

Dependent data; Local polynomial fitting; Nonparametric regression estimation; Strong consistency

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; CORRELATION METHODS; ERROR ANALYSIS; MATHEMATICAL MODELS; PARAMETER ESTIMATION; RANDOM PROCESSES; REGRESSION ANALYSIS;

KERNEL FUNCTIONS; NONPARAMETRIC ESTIMATION;

POLYNOMIALS;

EID: 0344861860     PISSN: 03610926     EISSN: None     Source Type: Journal    
DOI: 10.1081/STA-120025386     Document Type: Article

References (33)

1. Bradley, R. C. (1983). Approximation theorems for strongly mixing random variables. Michigan Math. J. 30:69-81.
2. Carbon, M., Tran, L. T., Wu, B. (1997). Kernel density estimation for random fields. Statist. Probab. Lett. 36:115-125.
3. Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 74:829-836.
4. Collomb, G., Härdie, W. (1986). Strong uniform convergence rates in robust nonparametric time series analysis and prediction: Kernel regression estimation from dependent observations. Stochastic Process. Appl. 23:77-89.
5. Fan, J. (1993). Local linear regression smoothers and their minimax efficiency. Ann. Statist. 21:196-216.
6. Fan, J., Gijbels, I. (1995). Data-driven bandwidth selection in local polynomial fitting: variable bandwidth and spatial adaptation. J. Roy. Statist. Soc. Ser. B 57(2):371-394.
7. Fan, J., Gijbels, I. (1996). Local Polynomial Modelling and its Applications. London: Chapman and Hall.
8. Francisco-Fernández, M., Vilar-Fernández, J. M. (2001). Local polynomial regression estimation with correlated errors. Comm. Statist. Theory and Methods 30(7):1271-1293.
9. Györfi, L., Härdle, W., Sarda, P., Vieu, P. (1989). Nonparametric curve estimation from time series. Lectures Notes in Statistics. Vol. 60. Berlin: Springer.
10. Hall, P., Heyde, C. C. (1980). Martingale Limit Theory and its Applications. New York: Academic Press.
11. Härdle, W., Tsybakov, A. (1997). Local polynomial estimators of the volatility function in nonparametric autoregression. J. Econometrics 81:223-242.
12. Hastie, T., Loader, C. (1993). Local regression: automatic kernel carpentry. Statist. Sci. 8(2):120-143.
13. Lu, Z., Cheng, P. (1997). Distribution-free strong consistency for non-parametric kernel regression involving nonlinear time series. J. Statist. Plann. Inference 65:67-86.
14. Masry, E. (1996a). Multivariate regression estimation: local polynomial fitting for time series. Stochastic Process. Appl. 65:81-101.
15. Masry, E. (1996b). Multivariate local polynomial regression for time series: uniform strong consistency and rates. J. Time Ser. Anal. 17:571-599.
16. Masry, E., Fan, J. (1997). Local polynomial estimation of regression function for mixing processes. Scand. J. Statist. 24:165-179.
17. Masry, E., Tjøstheim, D. (1995). Nonparametric estimation and identification of nonlinear ARCH time series. Econometric Theory 65:258-289.
18. Nze, P. A., Doukhan, P. (1998). Functional estimation for time series: uniform convergence properties. J. Statist. Plann. Inference 68:5-29.
19. Opsomer, J. D. (1997). Nonparametric regression in the presence of correlated errors. In: Gregoire, T. G., Brillinger, D. R., Diggle, P. J., Russek-Cohen, E., Warren, W. G., Wolfinger, R. D., eds. Modelling Longitudinal and Spatially Correlated Data: Methods, Applications and Future Directions. New York: Springer, pp. 339-348.
20. Pollard, D. (1984). Convergence of Stochastic Processes. New York: Springer.
21. Rio, E. (1995). The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann. Prob. 23(3):1188-1203.
22. Roussas, G. G. (1989). Consistent regression with fixed design points under dependence conditions. Statist. Probab. Lett. 8:41-50.
23. Roussas, G. G. (1990). Nonparametric regression estimation under mixing conditions. Stochastic Process. Appl. 36:107-116.
24. Roussas, G. G., Tran, L., Ionannides, D. (1992). Fixed design regression for time series: asymptotic normality. J. Multivariate Anal. 40(2):262-291.
25. Ruppert, D., Wand, P. (1994). Multivariate locally weighted least squares regression. Ann. Statist. 22:1346-1370.
26. Sacks, J., Ylvisaker, D. (1970). Designs for regression problems with correlated errors III. Ann. Math. Statist. 41:2057-2074.
27. Stone, C. J. (1977). Consistent nonparametric regression. Ann. Statist. 5:595-620.
28. Tran, L., Roussas, G., Yakowitz, S., Truong, V. (1996). Fixed-design regression for linear time series. Ann. Statist. 24(3):975-991.
29. Truong, Y. K., Stone, C. J. (1992). Nonparametric function estimation involving time series. Ann. Statist. 20:77-97.
30. Tsybakov, A. B. (1986). Robust reconstruction of functions by the local-approximation method. Problems Inform. Transmission 22:133-146.
31. Vilar-Fernández, J. A., Vilar-Fernández, J. M. (1998). Recursive estimation of regression functions by local polynomial fitting. Ann. Ins. Statist. Math. 50(4):729-754.
32. Vilar-Fernández, J. M., Francisco-Fernández, M. (2002). Local polynomial regression smoothers with AR-error structure. TEST 11(2):439-464.
33. Vilar-Fernández, J. M., Vilar-Fernández, J. A. (2000). Recursive local polynomial regression under dependence conditions. TEST 9(1):209-233.