Adaptive variance function estimation in heteroscedastic nonparametric regression

Annals of Statistics

Volumn 36, Issue 5, 2008, Pages 2025-2054

  Cai, T Tony ^a   Wang, Lie ^a  

^a UNIVERSITY OF PENNSYLVANIA   (United States)

Author keywords

Adaptive estimation; Nonparametric regression; Thresholding; Variance function estimation; Wavelets

Indexed keywords

EID: 54349089152     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/07-AOS509     Document Type: Article

References (23)

1. BERGER, J. (1976). Minimax estimation of a multivariate normal mean under arbitrary quadratic loss. J. Multivariate Anal. 4 642-648.
2. BROWN, L. D. and LEVINE, M. (2007). Variance estimation in nonparametric regression via the difference sequence method. Ann. Statist. 35 2219-2232.
3. BROWN, L. D. and LOW, M. G. (1996). A constrained risk inequality with applications to nonparametric functional estimations. Ann. Statist. 24 2524-2535.
4. CAI, T. (1999). Adaptive wavelet estimation: A block thresholding and oracle inequality approach. Ann. Statist. 27 898-924.
5. CAI, T. (2002). On block thresholding in wavelet regression: Adaptivity, block size, and threshold level. Statist. Sinica 12 1241-1273.
6. CAI, T. and SILVERMAN, B. W. (2001). Incorporating information on neighboring coefficients into wavelet estimation. Sankhyā Ser. B 63 127-148.
7. CAI, T. and WANG, L. (2007). Adaptive variance function estimation in heteroscedastic nonparametric regression. Technical report.
8. DAUBECHIES, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
9. DONOHO, D. L. and JOHNSTONE, I. M. (1994). Ideal spatial adaptation via wavelet shrinkage. Biometrika 81 425-455.
10. DONOHO, D. L., JOHNSTONE, I. M., KERKYACHARIAN, G. and PICARD, D. (1995). Wavelet shrinkage: Asymptopia? J. Roy. Statist. Soc. Ser. B 57 301-369.
11. FAN, J. and YAO, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85 645-660.
12. HALL, P. and CARROLL, R. J. (1989). Variance function estimation in regression: The effect of estimating the mean. J. Roy. Statist. Soc. Ser. B 51 3-14.
13. HALL, P., KERKYACHARIAN, G. and PICARD, D. (1998). Block threshold rules for curve estimation using kernel and wavelet methods. Ann. Statist. 26 922-942.
14. HALL, P., KERKYACHARIAN, G. and PICARD, D. (1999). On the minimax optimality of block thresholded wavelet estimators. Statist. Sinica 9 33-50.
15. HWANG, H. K. (1996). Large deviations for combinatorial distributions. I. Central limit theorems. Ann. Appl. Probab. 6 297-319.
16. JOHNSTONE, I. M. and SILVERMAN, B. W. (2005). Empirical Bayes selection of wavelet thresholds. Ann. Statist. 33 1700-1752.
17. LEPSKI, O. V. (1990). On a problem of adaptive estimation on white Gaussian noise. Theory Probab. Appl. 35 454-466.
18. MEYER, Y. (1992). Wavelets and Operators. Cambridge Univ. Press.
19. MÜLLER, H. G. and STADTMÜLLER, U. (1987). Estimation of heteroscedasticity in regression analysis. Ann. Statist. 15 610-625.
20. MÜLLER, H. G. and STADTMÜLLER, U. (1993). On variance function estimation with quadratic forms. J. Statist. Plann. Inference 35 213-231.
21. RUPPERT, D., WAND, M. P., HOLST, U. and HÖSSJER, O. (1997). Local polynomial variance function estimation. Technometrics 39 262-273.
22. TRIEBEL, H. (1983). Theory of Function Spaces. Birkhäuser, Basel.
23. WANG, L., BROWN, L. D., CAI, T. and LEVINE, M. (2008). Effect of mean on variance function estimation on nonparametric regression. Ann. Statist. 36 646-664.