Generalized Likelihood Ratio Tests for Varying-Coefficient Models with Censored Data

ABSTRACT

In this paper, we extend the generalized likelihood ratio test to the varying-coefficient models with censored data. We investigate the asymptotic behavior of the proposed test and demonstrate that its limiting null distribution follows a distribution, with the scale constant and the number of degree of freedom being independent of nuisance parameters or functions, which is called the wilks phenomenon. Both simulated and real data examples are given to illustrate the performance of the testing approach.

In this paper, we extend the generalized likelihood ratio test to the varying-coefficient models with censored data. We investigate the asymptotic behavior of the proposed test and demonstrate that its limiting null distribution follows a distribution, with the scale constant and the number of degree of freedom being independent of nuisance parameters or functions, which is called the wilks phenomenon. Both simulated and real data examples are given to illustrate the performance of the testing approach.

KEYWORDS

Varying Coefficient Model, Generalized Likelihood Ratio Test, Local Linear Method, Wilks Phenomenon, Censoring

Varying Coefficient Model, Generalized Likelihood Ratio Test, Local Linear Method, Wilks Phenomenon, Censoring

Cite this paper

nullR. Jiang and W. Qian, "Generalized Likelihood Ratio Tests for Varying-Coefficient Models with Censored Data,"*Open Journal of Statistics*, Vol. 1 No. 1, 2011, pp. 19-23. doi: 10.4236/ojs.2011.11003.

nullR. Jiang and W. Qian, "Generalized Likelihood Ratio Tests for Varying-Coefficient Models with Censored Data,"

References

[1] J. Buckley and I. R. James, "Linear Regression with Censored Data," Biometrika, Vol. 66, No. 3, 1979, pp. 429-436.

[2] Z. Cai, "Trending Time-Varying Coefficient Time Series Models with Serially Correlated Errors," Journal of Econometrics, Vol. 136, No. 1, 2007, pp. 163-188.

[3] K. Chen and X. W. Tong, "Varying Coefficient Transformation Models with Censored Data," Biometrika, Vol. 97, No. 4, 2010, pp. 969-976.

[4] D. M. Dabrowska, "Non-parametric Regression with Censored Survival Time Data," Scandinavian Journal of Statistics, Vol. 14, No. 3, 1987, pp. 181-197.

[5] J. Fan and I. Gijbels, "Censored Regression: Local Linear Approximations and Their Applications," Journal of the American Statistical Association, Vol. 89, No. 426, 1994, pp. 560-570.

[6] J. Fan and T. Huang, "Profile Likelihood Inferences on Semiparametric Varying-Coefficient Partially Linear Models," Bernolli, Vol. 11, No. 6, 2005, pp. 1031-1057.

[7] J. Fan, C. Zhang and J. Zhang, "Generalized Likelihood Ratio Statistics and Wilks Phenomenon," The Annals of Statistics, Vol. 29, No. 1, 2001, pp. 153-193.

[8] J. Fan and W. Zhang, "Statistical Estimation in Varying Coefficient Models," The Annals of Statistics, Vol. 27, No. 5, 1999, pp. 1491-1518.

[9] J. Fan and J. Zhang, "Two-Step Estimation of Functional Linear Models with Application to Longitudinal Data," Journal of Royal Statistical Association B, Vol. 62, No. 2, 2000, pp. 303-322.

[10] H. Wang and Y. Xia, "Shrinkage Estimation of The Varying Coefficient Model," Journal of the American Statistical Association, Vol. 104, No. 486, 2009, pp. 747-757.

[11] T. Hastie and R. Tibshirani, "Varying-Coefficient Models," Journal of Royal Statistical Association B, Vol. 55, No. 4, 1993, pp. 757-796.

[12] H. Koul, V. Susarla and J. Van Ryzin, "Regression Analysis with Randomly Right Censored Data," The Annals of Statistics, Vol. 9, No. 6, 1981, pp. 1276-1288.

[13] S. Leurgans, "Linear Models, Random Censoring and Synthetic Data," Biometrika, Vol. 74, No. 2, 1987, pp. 301-309.

[14] X. Luo, Z. Yang and Y. Zhou, "Varying-Coefficient Regression Models with Censored Data," Acta Mathematicae Applicatae Sinica, Vol. 29, No. 3, 2006, pp. 415-427.

[15] Z. Zheng, "A Class of Estimators of the Parameters in Linear Regression with Censored Data," Acta Mathematicae Applicatae Sinica, Vol. 3, No. 3, 1987, pp. 231-241.

[16] Z. Zheng, "Strong Consistency of Nonparametric Regression Estimates with Censored Data," Journal of Mathematical Research and Exposition, Vol. 8, No. 4, 1988, pp. 307-313.

[17] Y. Zhou and H. Liang, "Statistical Inference for Semiparametric Varying-Coefficient Partially Linear Models with Error-Prone Linear Covariates," The Annals of Statistics, Vol. 37, No. 1, 2009, pp. 427-458.

[1] J. Buckley and I. R. James, "Linear Regression with Censored Data," Biometrika, Vol. 66, No. 3, 1979, pp. 429-436.

[2] Z. Cai, "Trending Time-Varying Coefficient Time Series Models with Serially Correlated Errors," Journal of Econometrics, Vol. 136, No. 1, 2007, pp. 163-188.

[3] K. Chen and X. W. Tong, "Varying Coefficient Transformation Models with Censored Data," Biometrika, Vol. 97, No. 4, 2010, pp. 969-976.

[4] D. M. Dabrowska, "Non-parametric Regression with Censored Survival Time Data," Scandinavian Journal of Statistics, Vol. 14, No. 3, 1987, pp. 181-197.

[5] J. Fan and I. Gijbels, "Censored Regression: Local Linear Approximations and Their Applications," Journal of the American Statistical Association, Vol. 89, No. 426, 1994, pp. 560-570.

[6] J. Fan and T. Huang, "Profile Likelihood Inferences on Semiparametric Varying-Coefficient Partially Linear Models," Bernolli, Vol. 11, No. 6, 2005, pp. 1031-1057.

[7] J. Fan, C. Zhang and J. Zhang, "Generalized Likelihood Ratio Statistics and Wilks Phenomenon," The Annals of Statistics, Vol. 29, No. 1, 2001, pp. 153-193.

[8] J. Fan and W. Zhang, "Statistical Estimation in Varying Coefficient Models," The Annals of Statistics, Vol. 27, No. 5, 1999, pp. 1491-1518.

[9] J. Fan and J. Zhang, "Two-Step Estimation of Functional Linear Models with Application to Longitudinal Data," Journal of Royal Statistical Association B, Vol. 62, No. 2, 2000, pp. 303-322.

[10] H. Wang and Y. Xia, "Shrinkage Estimation of The Varying Coefficient Model," Journal of the American Statistical Association, Vol. 104, No. 486, 2009, pp. 747-757.

[11] T. Hastie and R. Tibshirani, "Varying-Coefficient Models," Journal of Royal Statistical Association B, Vol. 55, No. 4, 1993, pp. 757-796.

[12] H. Koul, V. Susarla and J. Van Ryzin, "Regression Analysis with Randomly Right Censored Data," The Annals of Statistics, Vol. 9, No. 6, 1981, pp. 1276-1288.

[13] S. Leurgans, "Linear Models, Random Censoring and Synthetic Data," Biometrika, Vol. 74, No. 2, 1987, pp. 301-309.

[14] X. Luo, Z. Yang and Y. Zhou, "Varying-Coefficient Regression Models with Censored Data," Acta Mathematicae Applicatae Sinica, Vol. 29, No. 3, 2006, pp. 415-427.

[15] Z. Zheng, "A Class of Estimators of the Parameters in Linear Regression with Censored Data," Acta Mathematicae Applicatae Sinica, Vol. 3, No. 3, 1987, pp. 231-241.

[16] Z. Zheng, "Strong Consistency of Nonparametric Regression Estimates with Censored Data," Journal of Mathematical Research and Exposition, Vol. 8, No. 4, 1988, pp. 307-313.

[17] Y. Zhou and H. Liang, "Statistical Inference for Semiparametric Varying-Coefficient Partially Linear Models with Error-Prone Linear Covariates," The Annals of Statistics, Vol. 37, No. 1, 2009, pp. 427-458.