Robust Inference for Time-Varying Coefficient Models with Longitudinal Data

Zhaofeng Wang; Jiancheng Jiang; Qunyi Qiu

doi:10.4236/ojs.2015.57070

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Open Journal of Statistics > Vol.5 No.7, December 2015

Robust Inference for Time-Varying Coefficient Models with Longitudinal Data ()

Zhaofeng Wang¹, Jiancheng Jiang², Qunyi Qiu³

¹Jishou University, Jishou, China.
²University of North Carolina, Charlotte, NC, USA.
³Peking University, Bejing, China.

DOI: 10.4236/ojs.2015.57070 PDF HTML XML 3,358 Downloads 3,945 Views

Abstract

Time-varying coefficient models are useful in longitudinal data analysis. Various efforts have been invested for the estimation of the coefficient functions, based on the least squares principle. Related work includes smoothing spline and kernel methods among others, but these methods suffer from the shortcoming of non-robustness. In this paper, we introduce a local M-estimation method for estimating the coefficient functions and develop a robustified generalized likelihood ratio (GLR) statistic to test if some of the coefficient functions are constants or of certain parametric forms. The robustified GLR test is robust against outliers and the error distribution. This provides a useful robust inference tool for the models with longitudinal data. The bandwidth selection issue is also addressed to facilitate the implementation in practice. Simulations show that the proposed testing method is more powerful in some situations than its counterpart based on the least squares principle. A real example is also given for illustration.

Keywords

Local Polynomial Smoothing, Longitudinal Data, Local M-Estimators, Generalized Likelihood Ratio Tests

Share and Cite:

Wang, Z. , Jiang, J. and Qiu, Q. (2015) Robust Inference for Time-Varying Coefficient Models with Longitudinal Data. Open Journal of Statistics, 5, 702-713. doi: 10.4236/ojs.2015.57070.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Diggle, P.J., Liang, K.-Y. and Zeger, S.L. (1994) Analysis of Longitudinal Data. Clarendon Press, Oxford.
[2]	Davidian, M. and Giltinan, D.M. (1995) Nonlinear Models for Repeated Measurement Data. Chapman and Hall, London.
[3]	Vonesh, E.F. and Chinchilli, V.M. (1997) Linear and Nonlinear Models for Analysis of Repeated Measurements. Marcel Dekker, New York.
[4]	Zeger, S.L. and Diggle, P.L. (1994) Semiparametric Models for Longitudinal Data with Application to CD4 Cell Numbers in HIV Seroconverters. Biometrics, 50, 689-699. http://dx.doi.org/10.2307/2532783
[5]	Müller, H.-G. (1998) Nonparametric Regression Analysis of Longitudinal Data. Lecture Notes in Statistics, Vol. 46, Springer-Verlag, Berlin.
[6]	Eubank, R.L., Huang, C., Maldonado, Y.M., Wang, N., Wang, S. and Buchanan, R.J. (2004) Smoothing Spline Estimation in Varying-Coefficient Models. Journal of the Royal Statistical Society, 66, 653-667. http://dx.doi.org/10.1111/j.1467-9868.2004.B5595.x
[7]	He, X., Fung, W.K. and Zhu, Z.Y. (2005) Robust Estimation in Generalized Partial Linear Models for Clustered Data. Journal of the American Statistical Association, 100, 1176-1184. http://dx.doi.org/10.1198/016214505000000277
[8]	He, X., Zhu, Z.Y. and Fung, W.K. (2002) Estimation in a Semiparametric Model for Longitudinal Data with Unspecified Dependence Structure. Biometrika, 89, 579-590. http://dx.doi.org/10.1093/biomet/89.3.579
[9]	Hoover, D.R., Rice, J.A., Wu, C.O. and Yang, L.-P. (1998) Nonparametric Smoothing Estimates of Time-Varing Coefficient Models with Longitudinal Data. Biometrika, 85, 809-822. http://dx.doi.org/10.1093/biomet/85.4.809
[10]	Fan, J. and 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. http://dx.doi.org/10.1111/1467-9868.00233
[11]	Huang, J., Wu, C. and Zhou, L. (2002) Vary-Coefficient Models and Basis Function Approximations for the Analysis of Repeated Measurements. Biometrika, 89, 111-128. http://dx.doi.org/10.1093/biomet/89.1.111
[12]	Lindsey, J.K. (1993) Models for Repeated Measurements. Oxford University Press, Oxford.
[13]	Jones, R.M. (1993) Longitudinal Data with Serial Correlation: A State-Space Approach. Chapman and Hall, London. http://dx.doi.org/10.1007/978-1-4899-4489-4
[14]	Hand, D. and Crower, M. (1996) Practical Longitudinal Data Analysis. Chapman and Hall, London. http://dx.doi.org/10.1007/978-1-4899-3033-0
[15]	Brumback, B. and Rice, J.A. (1988) Smoothing Spline Models for the Analysis of Nested and Crossed Samples of Curves (with Discussion). Journal of the American Statistical Association, 93, 961-994. http://dx.doi.org/10.1080/01621459.1998.10473755
[16]	Chiang, C.T., Rice, J.A. and Wu, C.O. (2001) Smoothing Spline Estimation for Varying Coefficient Models with Repeatedly Measured Dependent Variables. Journal of the American Statistical Association, 96, 605-619. http://dx.doi.org/10.1198/016214501753168280
[17]	Wu, C.O., Chiang, C.T. and Hoover, D.R. (1998) Asymptotic Confidence Regions for Kernal Smoothing of a Varying Coefficient Model with Longitudinal Data. Journal of the American Statistical Association, 93, 1388-1402. http://dx.doi.org/10.1080/01621459.1998.10473800
[18]	Wu, C.O. and Chiang, C.T. (2000) Kernal Smoothing on Varying Coefficient Models with Longitudinal Dependent Variable. Statistica Sinica, 10, 433-456.
[19]	Fan, J. and Jiang, J. (2000) Variable Bandwidth and One-Step Local M-Estimator. Science in China Series A, 43, 65-81. http://dx.doi.org/10.1007/BF02903849
[20]	Jiang, J. and Mack, Y.P. (2001) Robust Local Polynomial Regression for Dependent Data. Statistica Sinica, 11, 705-722.
[21]	Huber, P.J. (1964) Robust Estimation of a Location Parameter. The Annals of Mathematical Statistics, 35, 73-101. http://dx.doi.org/10.1214/aoms/1177703732
[22]	Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. Wiley, New York. http://dx.doi.org/10.1002/9780470316481
[23]	Jiang, J., Jiang, X. and Song, X. (2014) Weighted Composite Quantile Regression Estimation of DTARCH Models. Econometrics Journal, 17, 1-23. http://dx.doi.org/10.1111/ectj.12023
[24]	Hastie, T.J. and Tibshirani, R.J. (1990) Generalized Additive Models. Chapman and Hall, New York.
[25]	Fan, J., Zhang, C.-M. and Zhang, J. (2001) Generalized Likelihood Ratio Statistics and Wilks Phenomenon. The Annals of Statistics, 29, 153-193. http://dx.doi.org/10.1214/aos/996986505
[26]	Fan, J. and Huang, T. (2005) Profile Likelihood Inferences on Semiparametric Varying-Coefficient Partially Linear Models. Bernoulli, to Appear. http://dx.doi.org/10.3150/bj/1137421639
[27]	Fan, J. and Jiang, J. (2005) Nonparametric Inferences for Additive Models. Journal of the American Statistical Association, 100, 890-907. http://dx.doi.org/10.1198/016214504000001439
[28]	Fan, J. and Jiang, J. (2007) Nonparametric Inference with Generalized Likelihood Ratio Tests (with Discussion). Test, 16, 409-444. http://dx.doi.org/10.1007/s11749-007-0080-8
[29]	Hui, Y.V. and Jiang, J. (2005) Robust Modelling of DTARCH Models. The Econometrics Journal, 8, 143-158. http://dx.doi.org/10.1111/j.1368-423X.2005.00157.x
[30]	Brunner, E., Domhof, S. and Langer, F. (2002) Nonparametric Analysis of Longitudinal Data in Factorial Experiments. Wiley, New York.