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Adaptive varying-coefficient linear quantile model: a profiled estimating equations approach

  • Weihua Zhao
  • Jianbo Li
  • Heng Lian
Article
  • 260 Downloads

Abstract

We consider an estimating equations approach to parameter estimation in adaptive varying-coefficient linear quantile model. We propose estimating equations for the index vector of the model in which the unknown nonparametric functions are estimated by minimizing the check loss function, resulting in a profiled approach. The estimating equations have a bias-corrected form that makes undersmoothing of the nonparametric part unnecessary. The estimating equations approach makes it possible to obtain the estimates using a simple fixed-point algorithm. We establish asymptotic properties of the estimator using empirical process theory, with additional complication due to the nuisance nonparametric part. The finite sample performance of the new model is illustrated using simulation studies and a forest fire dataset.

Keywords

Asymptotic normality Bias-corrected estimating equations Check loss Empirical processes Single-index model 

Notes

Acknowledgements

We sincerely thank the AE and two anonymous reviewers for their insightful comments which have led to significant improvement of the paper. The research of Zhao was supported in part by National Social Science Fund of China (15BTJ027), and the research of Lian was supported by a start up Grant (No. 7200521/MA) from the City University of Hong Kong.

References

  1. Belloni, A., Chernozhukov, V. (2011). l1-penalized quantile regression in high-dimensional sparse models. The Annals of Statistics, 39, 82–130.Google Scholar
  2. Bondell, H. D., Reich, B. J., Wang, H. (2010). Noncrossing quantile regression curve estimation. Biometrika, 97, 825–838.Google Scholar
  3. Cai, Z., Xiao, Z. (2012). Semiparametric quantile regression estimation in dynamic models with partially varying coefficients. Journal of Econometrics, 167, 413–425.Google Scholar
  4. Carroll, R. J., Fan, J., Gijbels, I., Wand, M. P. (1997). Generalized partially linear single-index models. Journal of the American Statistical Association, 92, 477–489.Google Scholar
  5. Chen, R., Tsay, R. S. (1993). Functional-coefficient autoregressive models. Journal of the American Statistical Association, 88, 298–308.Google Scholar
  6. Cortez, P., Morais, A. (2007). A data mining approach to predict forest fires using meteorological data. In J. Neves, M. F. Santos and J. Machado (Eds.), New trends in artificial intelligence, Proceedings of the 13th EPIA 2007 - Portuguese Conference on Artificial Intelligence, 512–523.Google Scholar
  7. Cui, X., Haerdle, W. K., Zhu, L. (2011). The efm approach for single-index models. The Annals of Statistics, 39, 1658–1688.Google Scholar
  8. Fan, J., Fan, Y., Barut, E. (2014a). Adaptive robust variable selection. Annals of Statistics, 42, 324–351.Google Scholar
  9. Fan, J., Ma, Y., Dai, W. (2014b). Nonparametric independence screening in sparse ultra-high dimensional varying coefficient models. Journal of the American Statistical Association, 109, 1270–1284.Google Scholar
  10. Fan, J., Yao, Q., Cai, Z. (2003). Adaptive varying-coefficient linear models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65, 57–80.Google Scholar
  11. Fan, J. Q., Zhang, J. T. (2000). Two-step estimation of functional linear models with applications to longitudinal data. Journal of the Royal Statistical Society Series B-Statistical Methodology, 62, 303–322.Google Scholar
  12. Fan, J. Q., Zhang, W. Y. (1999). Statistical estimation in varying coefficient models. Annals of Statistics, 27, 1491–1518.Google Scholar
  13. Hall, P., Sheather, S. J. (1988). On the distribution of a studentized quantile. Journal of the Royal Statistical Society: Series B (Methodological), 50, 381–391.Google Scholar
  14. Hastie, T., Tibshirani, R. (1993). Varying-coefficient models. Journal of the Royal Statistical Society Series B-Methodological, 55, 757–796.Google Scholar
  15. He, X., Shi, P. (1994). Convergence rate of b-spline estimators of nonparametric conditional quantile functions. Journal of Nonparametric Statistics, 3, 299–308.Google Scholar
  16. Hendricks, W., Koenker, R. (1992). Hierarchical spline models for conditional quantiles and the demand for electricity. Journal of the American Statistical Association, 87, 58–68.Google Scholar
  17. Hoover, D. R., Rice, J. A., Wu, C. O., Yang, L. P. (1998). Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika, 85, 809–822.Google Scholar
  18. Horowitz, J. L., Lee, S. (2005). Nonparametric estimation of an additive quantile regression model. Journal of the American Statistical Association, 100, 1238–1249.Google Scholar
  19. Hu, Y., Gramacy, R., Lian, H. (2013). Bayesian quantile regression for single-index models. Statistics and Computing, 23, 437–454.Google Scholar
  20. Huang, J. H. Z., Wu, C. O., Zhou, L. (2002). Varying-coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika, 89, 111–128.Google Scholar
  21. Jiang, L., Wang, H. J., Bondell, H. D. (2013). Interquantile shrinkage in regression models. Journal of Computational and Graphical Statistics, 22, 970–986.Google Scholar
  22. Kim, M. (2007). Quantile regression with varying coefficients. Annals of Statistics, 35, 92–108.MathSciNetCrossRefzbMATHGoogle Scholar
  23. Koenker, R., Bassett, G, Jr. (1978). Regression quantiles. Econometrica: Journal of the Econometric Society, 1, 33–50.Google Scholar
  24. Koenker, R., Ng, P., Portnoy, S. (1994). Quantile smoothing splines. Biometrika, 81, 673–680.Google Scholar
  25. Kong, E., Xia, Y. (2012). A single-index quantile regression model and its estimation. Econometric Theory, 28, 730–768.Google Scholar
  26. Kottas, A., Krnjajic, M. (2009). Bayesian semiparametric modelling in quantile regression. Scandinavian Journal of Statistics, 36, 297–319.Google Scholar
  27. Lai, P., Wang, Q., Lian, H. (2012). Bias-corrected gee estimation and smooth-threshold gee variable selection for single-index models with clustered data. Journal of Multivariate Analysis, 105, 422–432.Google Scholar
  28. Lee, S. (2003). Efficient semiparametric estimation of a partially linear quantile regression model. Econometric Theory, 19, 1–31.MathSciNetCrossRefzbMATHGoogle Scholar
  29. Lian, H. (2012a). Semiparametric estimation of additive quantile regression models by two-fold penalty. Journal of Business & Economic Statistics, 30, 337–350.Google Scholar
  30. Lian, H. (2012b). Variable selection for high-dimensional generalized varying-coefficient models. Statistica Sinica, 22, 1563–1588.MathSciNetzbMATHGoogle Scholar
  31. Liu, J., Li, R., Wu, R. (2014). Feature selection for varying coefficient models with ultrahigh-dimensional covariates. Journal of the American Statistical Association, 109, 266–274.Google Scholar
  32. Lu, Z., Tjstheim, D., Yao, Q. (2007). Adaptive varying-coefficient linear models for stochastic processes: asymptotic theory. Statistica Sinica, 17, 177–198.Google Scholar
  33. Reich, B., Bondell, H., Wang, H. (2010). Flexible bayesian quantile regression for independent and clustered data. Biostatistics, 11, 337–352.Google Scholar
  34. Sherwood, B., Wang, L. (2016). Partially linear additive quantile regression in ultra-high dimension. Annals of Statistics, 44, 288–317.Google Scholar
  35. Tang, Y., Wang, H. J., Zhu, Z. (2013). Variable selection in quantile varying coefficient models with longitudinal data. Computational Statistics & Data Analysis, 57, 435–449.Google Scholar
  36. Tokdar, S., Kadane, J. B. (2011). Simultaneous linear quantile regression: A semiparametric bayesian approach. Bayesian Analysis, 6, 1–22.Google Scholar
  37. van der Geer, S. A. (2000). Empirical processes in M-estimation. Cambridge: Cambridge University Press.Google Scholar
  38. van der Vaart, A. W. (1998). Asymptotic statistics. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  39. van der Vaart, A. W., Wellner, J. A. (1996). Weak convergence and empirical processes. New York: Springer.Google Scholar
  40. Wang, H. J., Zhu, Z., Zhou, J. (2009). Quantile regression in partially linear varying coefficient models. The Annals of Statistics, 37, 3841–3866.Google Scholar
  41. Wang, J.-L., Xue, L., Zhu, L., Chong, Y. S. (2010). Estimation for a partial-linear single-index model. Annals of Statistics, 38, 246–274.Google Scholar
  42. Wang, L., Wu, Y., Li, R. (2012). Quantile regression for analyzing heterogeneity in ultra-high dimension. Journal of the American Statistical Association, 107, 214–222.Google Scholar
  43. Wei, F., Huang, J., Li, H. Z. (2011). Variable selection and estimation in high-dimensional varying-coefficient models. Statistica Sinica, 21, 1515–1540.Google Scholar
  44. Wu, T. Z., Yu, K., Yu, Y. (2010). Single-index quantile regression. Journal of Multivariate Analysis, 101, 1607–1621.Google Scholar
  45. Wu, Y., Liu, Y. (2009). Variable selection in quantile regression. Statistica Sinica, 19, 801–817.Google Scholar
  46. Xia, Y., Li, W. (1999). On single-index coefficient regression models. Journal of the American Statistical Association, 94, 1275–1285.Google Scholar
  47. Xue, L., Qu, A. (2012). Variable selection in high-dimensional varying-coefficient models with global optimality. The Journal of Machine Learning Research, 13, 1973–1998.Google Scholar
  48. Yang, Y., He, X. (2012). Bayesian empirical likelihood for quantile regression. The Annals of Statistics, 40, 1102–1131.Google Scholar
  49. Yu, K., Jones, M. (1998). Local linear quantile regression. Journal of the American Statistical Association, 93, 228–237.Google Scholar
  50. Yu, K., Moyeed, R. (2001). Bayesian quantile regression. Statistics & Probability Letters, 54, 437–447.Google Scholar
  51. Yu, Y., Ruppert, D. (2002). Penalized spline estimation for partially linear single-index models. Journal of the American Statistical Association, 97, 1042–1054.Google Scholar
  52. Zhu, L., Huang, M., Li, R. (2012). Semiparametric quantile regression with high-dimensional covariates. Statistica Sinica, 22, 1379–1401.Google Scholar
  53. Zhu, L., Lin, L., Cui, X., Li, G. (2010). Bias-corrected empirical likelihood in a multi-link semiparametric model. Journal of Multivariate Analysis, 101, 850–868.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2017

Authors and Affiliations

  1. 1.School of ScienceNantong UniversityNantongP. R. China
  2. 2.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouP. R. China
  3. 3.Department of MathematicsCity University of Hong KongKowloon TongHong Kong

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