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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 547874, 13 pages
http://dx.doi.org/10.1155/2013/547874
Research Article

Robust SiZer Approach for Varying Coefficient Models

Hui-Guo Zhang,1,2 Chang-Lin Mei,2 and He-Ling Wang3

1School of Mathematics and System Science, Xinjiang University, Urumqi 830000, China
2Department of Statistics, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
3School of Applied Mathematics, Xinjiang University of Finance and Economics, Urumqi 830000, China

Received 12 January 2013; Accepted 15 April 2013

Academic Editor: Joao B. R. Do Val

Copyright © 2013 Hui-Guo Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. S. Cleveland, E. Grosse, and M. J. Shyu, “Local regression models,” in Statistical Models in S. Pacific Grove, J. M. Chambers and T. Hastie, Eds., pp. 309–376, Wadsworth, Belmont, Calif, USA, 1992. View at Google Scholar
  2. T. Hastie and R. Tibshirani, “Varying-coefficient models,” Journal of the Royal Statistical Society B, vol. 55, no. 4, pp. 757–796, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C.-T. Chiang, J. A. Rice, and C. O. Wu, “Smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables,” Journal of the American Statistical Association, vol. 96, no. 454, pp. 605–619, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. L. Eubank, C. Huang, Y. Muñoz Maldonado, N. Wang, S. Wang, and R. J. Buchanan, “Smoothing spline estimation in varying-coefficient models,” Journal of the Royal Statistical Society B, vol. 66, no. 3, pp. 653–667, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Fan and W. Zhang, “Statistical estimation in varying coefficient models,” The Annals of Statistics, vol. 27, no. 5, pp. 1491–1518, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. Fan and J.-T. Zhang, “Two-step estimation of functional linear models with applications to longitudinal data,” Journal of the Royal Statistical Society B, vol. 62, no. 2, pp. 303–322, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  7. Z. Cai, J. Fan, and R. Li, “Efficient estimation and inferences for varying-coefficient models,” Journal of the American Statistical Association, vol. 95, no. 451, pp. 888–902, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Z. Cai, “Two-step likelihood estimation procedure for varying-coefficient models,” Journal of Multivariate Analysis, vol. 82, no. 1, pp. 189–209, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. Fan and W. Zhang, “Simultaneous confidence bands and hypothesis testing in varying-coefficient models,” Scandinavian Journal of Statistics, vol. 27, no. 4, pp. 715–731, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. W. Zhang and H. Peng, “Simultaneous confidence band and hypothesis test in generalised varying-coefficient models,” Journal of Multivariate Analysis, vol. 101, no. 7, pp. 1656–1680, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. G. Zhang and C. L. Mei, “Local least absolute deviation estimation of spatially varying coefficient models: robust geographically weighted regression approaches,” International Journal of Geographical Information Science, vol. 25, pp. 1467–1489, 2011. View at Google Scholar
  12. Q. Tang and J. Wang, “L1-estimation for varying coefficient models,” Statistics, vol. 39, no. 5, pp. 389–404, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. T. Qingguo and C. Longsheng, “M-estimation and B-spline approximation for varying coefficient models with longitudinal data,” Journal of Nonparametric Statistics, vol. 20, no. 7, pp. 611–625, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. T. Qingguo and C. Longsheng, “Asymptotic normality of M-estimators for varying coefficient models with longitudinal data,” Communications in Statistics, vol. 38, no. 8–10, pp. 1422–1440, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  15. T. Honda, “Quantile regression in varying coefficient models,” Journal of Statistical Planning and Inference, vol. 121, no. 1, pp. 113–125, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M.-O. Kim, “Quantile regression with varying coefficients,” The Annals of Statistics, vol. 35, no. 1, pp. 92–108, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Z. Cai and X. Xu, “Nonparametric quantile estimations for dynamic smooth coefficient models,” Journal of the American Statistical Association, vol. 104, no. 485, pp. 371–383, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A. Azzalini and A. W. Bowman, “On the use of nonparametric regression for checking linear relationships,” Journal of the Royal Statistical Society B, vol. 55, no. 2, pp. 549–557, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. W. González-Manteiga, M. D. Martínez-Miranda, and R. Raya-Miranda, “SiZer map for inference with additive models,” Statistics and Computing, vol. 18, no. 3, pp. 297–312, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  20. P. Chaudhuri and J. S. Marron, “SiZer for exploration of structures in curves,” Journal of the American Statistical Association, vol. 94, no. 447, pp. 807–823, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. P. Chaudhuri and J. S. Marron, “Scale space view of curve estimation,” The Annals of Statistics, vol. 28, no. 2, pp. 408–428, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J. Hannig and J. S. Marron, “Advanced distribution theory for SiZer,” Journal of the American Statistical Association, vol. 101, no. 474, pp. 484–499, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. P. Erästö and L. Holmström, “Bayesian multiscale smoothing for making inferences about features in scatterplots,” Journal of Computational and Graphical Statistics, vol. 14, no. 3, pp. 569–589, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  24. F. Godtliebsen and T. A. Øigård, “A visual display device for significant features in complicated signals,” Computational Statistics & Data Analysis, vol. 48, no. 2, pp. 317–343, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  25. T. A. Øigård, H. Rue, and F. Godtliebsen, “Bayesian multiscale analysis for time series data,” Computational Statistics & Data Analysis, vol. 51, no. 3, pp. 1719–1730, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  26. B. Ganguli and M. P. Wand, “Feature significance in geostatistics,” Journal of Computational and Graphical Statistics, vol. 13, no. 4, pp. 954–973, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  27. F. Godtliebsen, J. S. Marron, and P. Chaudhuri, “Significance in scale space for bivariate density estimation,” Journal of Computational and Graphical Statistics, vol. 11, no. 1, pp. 1–21, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  28. F. Godtliebsen, J. S. Marron, and P. Chaudhuri, “Statistical significance of features in digital images,” Image and Vision Computing, vol. 22, pp. 1093–1104, 2004. View at Google Scholar
  29. J. S. Marron and J.-T. Zhang, “SiZer for smoothing splines,” Computational Statistics, vol. 20, no. 3, pp. 481–502, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. C. Park and K.-H. Kang, “SiZer analysis for the comparison of regression curves,” Computational Statistics & Data Analysis, vol. 52, no. 8, pp. 3954–3970, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  31. C. Park, J. Hannig, and K.-H. Kang, “Improved SiZer for time series,” Statistica Sinica, vol. 19, no. 4, pp. 1511–1530, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. C. Park, T. C. M. Lee, and J. Hannig, “Multiscale exploratory analysis of regression quantiles using quantile SiZer,” Journal of Computational and Graphical Statistics, vol. 19, no. 3, pp. 497–513, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  33. L. Holmström, L. Pasanen, R. Furrer, and S. R. Sain, “Scale space multiresolution analysis of random signals,” Computational Statistics & Data Analysis, vol. 55, no. 10, pp. 2840–2855, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. A. Vaughan, M. Jun, and C. Park, “Statistical inference and visualization in scale-space for spatially dependent images,” Journal of the Korean Statistical Society, vol. 41, no. 1, pp. 115–135, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  35. H.-G. Zhang and C.-L. Mei, “SiZer inference for varying coefficient models,” Communications in Statistics, vol. 41, no. 10, pp. 1944–1959, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  36. J. Hannig and T. C. M. Lee, “Robust SiZer for exploration of regression structures and outlier detection,” Journal of Computational and Graphical Statistics, vol. 15, no. 1, pp. 101–117, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  37. F. T. Wang and D. W. Scott, “The L1 method for robust nonparametric regression,” Journal of the American Statistical Association, vol. 89, no. 425, pp. 65–76, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. H. M. Wagner, “Linear programming techniques for regression analysis,” Journal of the American Statistical Association, vol. 54, pp. 206–212, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. R. Koenker, Quantile Regression, Cambridge University Press, Cambridge, UK, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  40. J. L. Horowitz, “Bootstrap methods for median regression models,” Econometrica, vol. 66, no. 6, pp. 1327–1351, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. M. Buchinsky, “Estimating the asymptotic covariance matrix for quantile regression models: a Monte Carlo study,” Joumal of Econometrics, vol. 68, pp. 303–338, 1995. View at Google Scholar
  42. D. De Angelis, P. Hall, and G. A. Young, “Analytical and bootstrap approximations to estimator distributions in L1 regression,” Journal of the American Statistical Association, vol. 88, no. 424, pp. 1310–1316, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. Y. Sun, “A consistent nonparametric equality test of conditional quantile functions,” Econometric Theory, vol. 22, no. 4, pp. 614–632, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. X. Feng, X. He, and J. Hu, “Wild bootstrap for quantile regression,” Biometrika, vol. 98, no. 4, pp. 995–999, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. R. Cao-Abad, “Rate of convergence for the wild bootstrap in nonparametric regression,” The Annals of Statistics, vol. 19, no. 4, pp. 2226–2231, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet