Abstract
This paper is concerned with the heteroscedastic regression model Y i=g(x i)+σ iei(1⩽i⩽n) under correlated errors e i, where it is assumed that σ Emphasis>/2 i =f(u i), the design points (x i, ui) are known and nonrandom, and g and f are unknown functions. Assuming that unobserved disturbances e i are martingale differences. The strong uniform convergence rates and r-th moment uniform convergence rates of wavelet estimator of g are investigated. Also, the strong uniform convergence rates are discussed for wavelet estimator of f.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Antoniadis, A., Gregoire, G., Mckeague, I. W., Wavelet methods for curve estimation, J. Amer. Statist. Assoc., 1994, 89:1340–1352.
Chen, M. H., Ren, Z., Hu, S. H., Strong consistency of a class of estimators in partial linear model, Acta Math. Sinica, 1998, 41(2):429–439.
Chen, Z. J., Liang, H. Y., Ren, Y. F., Strong consistency of estimator in heteroscedastic model under NA samples, J. Tongji Univ., Nat. Sci., 2003, 31(8):1001–1005.
Donoho, D. L., Johnston, I. M., Kerkyacharian, G., et al., Density estimation by wavelet thresholding, Ann. Statist., 1996, 24:508–539.
Gao, J. T., Chen, X. R., Zhao, L. C., Asymptotic normality of a class of estimators in partial linear models, Acta Math. Sinica, 1994, 37(2):256–268.
Georgiev, A. A., Consistent nonparametric multiple regression: the fixed design case, J. Multivariate Anal., 1988, 25(1):100–110.
Hall, P., Heyde, C. C., Martingale Limit Theory and Its Applications, New York: Academic Press, 1980.
Mitrinovic, D. S., Analytic Inequalities, New York:Springer, 1970.
Priestley, M. B., Chao, M. T., Nonparametric function fitting, J. Roy. Statist. Soc. Ser. B, 1972, 34:385–392.
Qian, W. M., Chai, G. X., Jiang, F. Y., Wavelet estimation of error variance in semiparametric regression model, Chinese Ann. Math. Ser. A, 2000, 21(3):341–350.
Roussas, G. G., Consistent regression with fixed design points under dependence condition, Statist. Probab. Lett., 1989, 8:41–50.
Roussas, G. G., Tran, L. T., Ioannides, D. A., Fixed design regression for time series: asymptotic normality, J. Multivariate Anal., 1992, 40:262–291.
Wang, Q. H., Some convergence properties of weighted kernel estimators of regression function in nonparametric model under random censorship, Acta Math. Appl. Sinica, 1996, 19(3):338–350.
Watler, G. G., Wavelets and Orthogonal Systems with Applications, Florida:CRC Press, Inc., 1994.
Xue, L. G., Strong uniform convergnece rates of the wavelet estimator of regression function under completed and censored data, Acta Math. Appl. Sinica, 2002a, 25(3):430–438.
Xue, L. G., Uniform convergence rates of the wavelet estimator of regression function under mixing error, Acta Math. Sci., Ser. A, 2002b, 22(4):528–535.
Author information
Authors and Affiliations
Additional information
Partially supported by the National Natural Science Foundation of China (10171079).
Rights and permissions
About this article
Cite this article
Liang, H., Zhang, D. & Lu, B. Wavelet estimation in nonparametric model under martingale difference errors. Appl. Math. Chin. Univ. 19, 302–310 (2004). https://doi.org/10.1007/s11766-004-0039-4
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11766-004-0039-4