Liang, Han-Ying;Qi, Yan-Yan

  • Published : 2007.05.31


Consider the heteroscedastic regression model $Y_i=g(x_i)+{\sigma}_i\;{\epsilon}_i=(1{\leq}i{\leq}n)$, where ${\sigma}^2_i=f(u_i)$, the design points $(x_i,\;u_i)$ are known and nonrandom, and g and f are unknown functions defined on closed interval [0, 1]. Under the random errors $\epsilon_i$ form a sequence of NA random variables, we study the asymptotic normality of wavelet estimators of g when f is a known or unknown function.


regression function;NA error;wavelet estimator;asymptotic normality


  1. K. Alam and K. M. L. Saxena, Positive dependence in multivariate distributions, Comm. Statist. A-Theory Methods 10 (1981), no. 12, 1183-1196
  2. A. Antoniadis, G. Gregoire, and I. W. McKeague, Wavelet methods for curve estimation, J. Amer. Statist. Assoc. 89 (1994), no. 428, 1340-1353
  3. J.-I. Baek, T.-S. Kim, and H.-Y. Liang, On the convergence of moving average processes under dependent conditions, Aust. N. Z. J. Stat. 45 (2003), no. 3, 331-342
  4. M. H. Chen, Z. Ren, and S. Hu, Strong consistency of a class of estimators in a partial linear model, Acta Math. Sinica (Chin. Ser.) 41 (1998), no. 2, 429-438
  5. Z. J. Chen, H. Y. Liang, and Y. F. Ren, Strong consistency of estimators in a heteroscedastic model under NA samples, Tongji Daxue Xuebao Ziran Kexue Ban 31 (2003), no. 8, 1001-1005
  6. D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, Density estimation by wavelet thresholding, Ann. Statist. 24 (1996), no. 2, 508-539
  7. T. Gasser and H. Muller, Smoothing techniques for curve estimation, Proceedings of a Workshop held in Heidelberg, April 2-4, 1979. Edited by Th. Gasser and M. Rosenblatt. Lecture Notes in Mathematics, 757. Springer, Berlin, 1979
  8. A. A. Georgiev, Consistent nonparametric multiple regression: the fixed design case, J. Multivariate Anal. 25 (1988), no. 1, 100-110
  9. P. Hall and P. Patil, On wavelet methods for estimating smooth functions, Bernoulli 1 (1995), no. 1-2, 41-58
  10. H.-Y. Liang, Complete convergence for weighted sums of negatively associated random variables, Statist. Probab. Lett. 48 (2000), no. 4, 317-325
  11. H.-Y. Liang and J.-I. Baek, Weighted sums of negatively associated random variables, Aust. N. Z. J. Stat. 48 (2006), no. 1, 21-31
  12. H.-Y. Liang, D. Zhang, and B. Lu, Wavelet estimation in nonparametric model under martingale difference errors, Appl. Math. J. Chinese Univ. Ser. B 19 (2004), no. 3, 302-310
  13. H. Liang, L. Zhu, and Y. Zhou, Asymptotically efficient estimation based on wavelet of expectation value in a partial linear model, Comm. Statist. Theory Methods 28 (1999), no. 9, 2045-2055
  14. D. S. Mitrinovic, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970
  15. W. M. Qian and G. X. Cai, The strong convergencerate of wavelet estimator in partially models, Chinese Sci. A29 (1999), 233-240
  16. W. M. Qian, G. X. Chai, and F. Y. Jiang, The wavelet estimation of error variance for semiparametric regression models, Chinese Ann. Math. Ser. A 21 (2000), no. 3, 341-350
  17. G. G. Roussas, Consistent regression estimation with fixed design points under dependence conditions, Statist. Probab. Lett. 8 (1989), no. 1, 41-50
  18. G. G. Roussas, Asymptotic normality of random fields of positively or negatively associated processes, J. Multivariate Anal. 50 (1994), no. 1, 152-173
  19. G. G. Roussas, L. T. Tran, and D. A. Ioannides, Fixed design regression for time series: asymptotic normality, J. Multivariate Anal. 40 (1992), no. 2, 262-291
  20. Q.-M. Shao, A comparison theorem on moment inequalities between negatively associated and independent random variables, J. Theoret. Probab. 13 (2000), no. 2, 343-356
  21. Q. H. Wang, Some convergence properties of weighted kernel estimators of regression functions under random censorship, Acta Math. Appl. Sinica 19 (1996), no. 3, 338-350
  22. G. G. Walter, Wavelets and other orthogonal systems with applications, CRC Press, Boca Raton, FL, 1994
  23. L. G. Xue, Strong uniform convergence rates of wavelet estimates of regression function under complete and censored data, Acta Math. Appl. Sin. 25 (2002), no. 3, 430-438
  24. X. Zhou and J. You, Wavelet estimation in varying-coefficient partially linear regression models, Statist. Probab. Lett. 68 (2004), no. 1, 91-104
  25. Z. Cai and G. G. Roussas, Berry-Esseen bounds for smooth estimator of a distribution function under association, First NIU Symposium on Statistical Sciences (De Kalb, IL, 1996). J. Nonparametr. Statist. 11 (1999), no. 1-3, 79-106
  26. K. Joag-Dev and F. Proschan, Negative association of random variables, with applications, Ann. Statist. 11 (1983), no. 1, 286-295
  27. H.-Y. Liang and B.-Y. Jing, Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences, J. Multivariate Anal. 95 (2005), no. 2, 227-245
  28. G. G. Roussas, Asymptotic normality of the kernel estimate of a probability density function under association, Statist. Probab. Lett. 50 (2000), no. 1, 1-12
  29. Q.-M. Shao and C. Su, The law of the iterated logarithm for negatively associated random variables, Stochastic Process. Appl. 83 (1999), no. 1, 139-148

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