DOI QR코드

DOI QR Code

ASYMPTOTIC NORMALITY OF WAVELET ESTIMATOR OF REGRESSION FUNCTION UNDER NA ASSUMPTIONS

  • Published : 2007.05.31

Abstract

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.

Keywords

References

  1. K. Alam and K. M. L. Saxena, Positive dependence in multivariate distributions, Comm. Statist. A-Theory Methods 10 (1981), no. 12, 1183-1196 https://doi.org/10.1080/03610928108828102
  2. A. Antoniadis, G. Gregoire, and I. W. McKeague, Wavelet methods for curve estimation, J. Amer. Statist. Assoc. 89 (1994), no. 428, 1340-1353 https://doi.org/10.2307/2290996
  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 https://doi.org/10.1111/1467-842X.00287
  4. 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 https://doi.org/10.1080/10485259908832776
  5. 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
  6. 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
  7. D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, Density estimation by wavelet thresholding, Ann. Statist. 24 (1996), no. 2, 508-539 https://doi.org/10.1214/aos/1032894451
  8. 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
  9. A. A. Georgiev, Consistent nonparametric multiple regression: the fixed design case, J. Multivariate Anal. 25 (1988), no. 1, 100-110 https://doi.org/10.1016/0047-259X(88)90155-8
  10. P. Hall and P. Patil, On wavelet methods for estimating smooth functions, Bernoulli 1 (1995), no. 1-2, 41-58 https://doi.org/10.2307/3318680
  11. K. Joag-Dev and F. Proschan, Negative association of random variables, with applications, Ann. Statist. 11 (1983), no. 1, 286-295 https://doi.org/10.1214/aos/1176346079
  12. H.-Y. Liang, Complete convergence for weighted sums of negatively associated random variables, Statist. Probab. Lett. 48 (2000), no. 4, 317-325 https://doi.org/10.1016/S0167-7152(00)00002-X
  13. 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 https://doi.org/10.1111/j.1467-842X.2006.00422.x
  14. 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 https://doi.org/10.1016/j.jmva.2004.06.004
  15. 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 https://doi.org/10.1007/s11766-004-0039-4
  16. 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 https://doi.org/10.1080/03610929908832405
  17. D. S. Mitrinovic, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970
  18. W. M. Qian and G. X. Cai, The strong convergencerate of wavelet estimator in partially models, Chinese Sci. A29 (1999), 233-240
  19. 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
  20. G. G. Roussas, Consistent regression estimation with fixed design points under dependence conditions, Statist. Probab. Lett. 8 (1989), no. 1, 41-50 https://doi.org/10.1016/0167-7152(89)90081-3
  21. G. G. Roussas, Asymptotic normality of random fields of positively or negatively associated processes, J. Multivariate Anal. 50 (1994), no. 1, 152-173 https://doi.org/10.1006/jmva.1994.1039
  22. 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 https://doi.org/10.1016/S0167-7152(00)00072-9
  23. 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 https://doi.org/10.1016/0047-259X(92)90026-C
  24. 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 https://doi.org/10.1023/A:1007849609234
  25. 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 https://doi.org/10.1016/S0304-4149(99)00026-5
  26. 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
  27. G. G. Walter, Wavelets and other orthogonal systems with applications, CRC Press, Boca Raton, FL, 1994
  28. 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
  29. X. Zhou and J. You, Wavelet estimation in varying-coefficient partially linear regression models, Statist. Probab. Lett. 68 (2004), no. 1, 91-104 https://doi.org/10.1016/j.spl.2004.01.018

Cited by

  1. Berry–Esseen bounds for wavelet estimator in a regression model with linear process errors vol.81, pp.1, 2011, https://doi.org/10.1016/j.spl.2010.09.024
  2. Asymptotic normality of wavelet estimator in heteroscedastic model with α-mixing errors vol.24, pp.4, 2011, https://doi.org/10.1007/s11424-010-8354-8
  3. CLT of wavelet estimator in semiparametric model with correlated errors vol.25, pp.3, 2012, https://doi.org/10.1007/s11424-012-0166-6
  4. Berry–Esseen type bounds of estimators in a semiparametric model with linear process errors vol.100, pp.1, 2009, https://doi.org/10.1016/j.jmva.2008.03.006
  5. Asymptotic normality of wavelet estimator for strong mixing errors vol.38, pp.4, 2009, https://doi.org/10.1016/j.jkss.2009.03.002
  6. Asymptotic properties of wavelet estimators in semiparametric regression models under dependent errors vol.122, 2013, https://doi.org/10.1016/j.jmva.2013.08.006
  7. The Berry-Esseen bounds of wavelet estimator for regression model whose errors form a linear process with a ρ-mixing vol.2016, pp.1, 2016, https://doi.org/10.1186/s13660-016-1036-x
  8. A Berry-Esseen Type Bound of Wavelet Estimator Under Linear Process Errors Based on a Strong Mixing Sequence vol.42, pp.22, 2013, https://doi.org/10.1080/03610926.2011.642921
  9. Asymptotics of a wavelet estimator in the nonparametric regression model with repeated measurements under a NA error process vol.109, pp.1, 2015, https://doi.org/10.1007/s13398-014-0172-8
  10. The Recursive Kernel Distribution Function Estimator Based on Negatively and Positively Associated Sequences vol.39, pp.20, 2010, https://doi.org/10.1080/03610920903427750
  11. Asymptotic normality of variance estimator in a heteroscedastic model with dependent errors vol.23, pp.2, 2011, https://doi.org/10.1080/10485252.2011.552721