Empirical Likelihood Diagnosis of Modal Linear Regression Models
Affiliation(s)
1 Department of Fundamental Course, Air Force Logistics College, Xuzhou, China. 2 School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, China. 3 Fundamental Science Department, North China Institute of Astronautic Engineering, Langfang, China.
1 Department of Fundamental Course, Air Force Logistics College, Xuzhou, China. 2 School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, China. 3 Fundamental Science Department, North China Institute of Astronautic Engineering, Langfang, China.
ABSTRACT
In this paper, we investigate the empirical likelihood diagnosis of modal linear regression models. The empirical likelihood ratio function based on modal regression estimation method for the regression coefficient is introduced. First, the estimation equation based on empirical likelihood method is established. Then, some diagnostic statistics are proposed. At last, we also examine the performance of proposed method for finite sample sizes through simulation study.
In this paper, we investigate the empirical likelihood diagnosis of modal linear regression models. The empirical likelihood ratio function based on modal regression estimation method for the regression coefficient is introduced. First, the estimation equation based on empirical likelihood method is established. Then, some diagnostic statistics are proposed. At last, we also examine the performance of proposed method for finite sample sizes through simulation study.
Cite this paper
Wang, S. , Zheng, L. and Dai, J. (2014) Empirical Likelihood Diagnosis of Modal Linear Regression Models. Journal of Applied Mathematics and Physics, 2, 948-952. doi: 10.4236/jamp.2014.210107.
Wang, S. , Zheng, L. and Dai, J. (2014) Empirical Likelihood Diagnosis of Modal Linear Regression Models. Journal of Applied Mathematics and Physics, 2, 948-952. doi: 10.4236/jamp.2014.210107.
References
[1] Muller, D.W. and Sawitzki, G. (1991) Excess Mass Estimates and Tests for Multimodality. Journal of the American Statistical Association, 86, 738-746. [2] Scott, D.W. (1992) Multivariate Density Estimation: Theory, Practice and Visualization. Wiley, New York.
http://dx.doi.org/10.1002/9780470316849 [3] Friedman, J.H. and Fisher, N.I. (1999) Bump Hunting in High-Dimensional Data. Statistics and Computing, 9, 123-143.
http://dx.doi.org/10.1023/A:1008894516817 [4] Chaudhuri, P. and Marron, J.S. (1999) Sizer for Exploration of Structures in Curves. Journal of the American Statistical Association, 94, 807-823. http://dx.doi.org/10.1080/01621459.1999.10474186 [5] Fisher, N.I. and Marron, J.S. (2001) Mode Testing via the Excess Mass Estimate. Biometrika, 88, 499-517.
http://dx.doi.org/10.1093/biomet/88.2.499 [6] Davies, P.L. and Kovac, A. (2004) Densities, Spectral Densities and Modality. Annals of Statistics, 32, 1093-1136. [7] Hall, P., Minnotte, M.C. and Zhang, C. (2004) Bump Hunting with Non-Gaussian Kernels. Annals of Statistics, 32, 2124-2141. http://dx.doi.org/10.1214/009053604000000715 [8] Ray, S. and Lindsay, B.G. (2005) The Topography of Multivariate Normal Mixtures. Annals of Statistics, 33, 2042-2065. http://dx.doi.org/10.1214/009053605000000417 [9] Yao, W. and Lindsay, B.G. (2009) Bayesian Mixture Labeling by Highest Posterior Density. Journal of American Statistical Association, 104, 758-767. http://dx.doi.org/10.1198/jasa.2009.0237 [10] Yao, W. and Li, L. (2013) A New Regression Model: Modal Linear Regression. Scandinavian Journal of Statistics, 41, 656-671. http://dx.doi.org/10.1111/sjos.12054 [11] Lee, M.J. (1989) Mode Regression. Journal of Econometrics, 42, 337-349.
http://dx.doi.org/10.1016/0304-4076(89)90057-2 [12] Yao, W., Lindsay, B. and Li, R. (2012) Local Modal Regression. Journal of Nonparametric Statistics, 24, 647-663.
http://dx.doi.org/10.1080/10485252.2012.678848 [13] Yu, K. and Aristodemou, K. (2012) Bayesian Mode Regression. Technical Report. arXiv: 1208.0579v1. [14] Zhao, W.H., Zhang, R.Q., Liu, Y.K. and Liu, J.C. (2014) Empirical Likelihood Based Modal Regression. Statistical Papers. [15] Thomas, D.R. and Grunkemeier, G.L. (1975) Confidence Interval Estimation of Survival Interval Estimation of Survival Probabilities for Censored Data. Journal of the American Statistical Association, 70, 865-871.
http://dx.doi.org/10.1080/01621459.1975.10480315 [16] Owen, A. (2001) Empirical Likelihood. Chapman and Hall, New York. ，
http://dx.doi.org/10.1201/9781420036152 [17] Zhu, H.T., Ibrahim, J.G., Tang, N.S. and Zhang, H.P. (2008) Diagnostic Measures for Empirical Likelihood of Generalized Estimating Equations. Biometrika, 95, 489-507. http://dx.doi.org/10.1093/biomet/asm094 [18] Xue, L.G. and Zhu, L.X. (2010) Empirical Likelihood in Nonparametric and Semiparametric Models. Science Press, Beijing. [19] Cook, R.D. and Weisberg, S. (1982) Residuals and Influence in Regression. Chapman and Hall, New York. [20] Wei, B.C., Lu, G.B. and Shi, J.Q. (1990) Statistical Diagnostics. Publishing House of Southeast University, Nanjing.
[1] Muller, D.W. and Sawitzki, G. (1991) Excess Mass Estimates and Tests for Multimodality. Journal of the American Statistical Association, 86, 738-746. [2] Scott, D.W. (1992) Multivariate Density Estimation: Theory, Practice and Visualization. Wiley, New York.
http://dx.doi.org/10.1002/9780470316849 [3] Friedman, J.H. and Fisher, N.I. (1999) Bump Hunting in High-Dimensional Data. Statistics and Computing, 9, 123-143.
http://dx.doi.org/10.1023/A:1008894516817 [4] Chaudhuri, P. and Marron, J.S. (1999) Sizer for Exploration of Structures in Curves. Journal of the American Statistical Association, 94, 807-823. http://dx.doi.org/10.1080/01621459.1999.10474186 [5] Fisher, N.I. and Marron, J.S. (2001) Mode Testing via the Excess Mass Estimate. Biometrika, 88, 499-517.
http://dx.doi.org/10.1093/biomet/88.2.499 [6] Davies, P.L. and Kovac, A. (2004) Densities, Spectral Densities and Modality. Annals of Statistics, 32, 1093-1136. [7] Hall, P., Minnotte, M.C. and Zhang, C. (2004) Bump Hunting with Non-Gaussian Kernels. Annals of Statistics, 32, 2124-2141. http://dx.doi.org/10.1214/009053604000000715 [8] Ray, S. and Lindsay, B.G. (2005) The Topography of Multivariate Normal Mixtures. Annals of Statistics, 33, 2042-2065. http://dx.doi.org/10.1214/009053605000000417 [9] Yao, W. and Lindsay, B.G. (2009) Bayesian Mixture Labeling by Highest Posterior Density. Journal of American Statistical Association, 104, 758-767. http://dx.doi.org/10.1198/jasa.2009.0237 [10] Yao, W. and Li, L. (2013) A New Regression Model: Modal Linear Regression. Scandinavian Journal of Statistics, 41, 656-671. http://dx.doi.org/10.1111/sjos.12054 [11] Lee, M.J. (1989) Mode Regression. Journal of Econometrics, 42, 337-349.
http://dx.doi.org/10.1016/0304-4076(89)90057-2 [12] Yao, W., Lindsay, B. and Li, R. (2012) Local Modal Regression. Journal of Nonparametric Statistics, 24, 647-663.
http://dx.doi.org/10.1080/10485252.2012.678848 [13] Yu, K. and Aristodemou, K. (2012) Bayesian Mode Regression. Technical Report. arXiv: 1208.0579v1. [14] Zhao, W.H., Zhang, R.Q., Liu, Y.K. and Liu, J.C. (2014) Empirical Likelihood Based Modal Regression. Statistical Papers. [15] Thomas, D.R. and Grunkemeier, G.L. (1975) Confidence Interval Estimation of Survival Interval Estimation of Survival Probabilities for Censored Data. Journal of the American Statistical Association, 70, 865-871.
http://dx.doi.org/10.1080/01621459.1975.10480315 [16] Owen, A. (2001) Empirical Likelihood. Chapman and Hall, New York. ，
http://dx.doi.org/10.1201/9781420036152 [17] Zhu, H.T., Ibrahim, J.G., Tang, N.S. and Zhang, H.P. (2008) Diagnostic Measures for Empirical Likelihood of Generalized Estimating Equations. Biometrika, 95, 489-507. http://dx.doi.org/10.1093/biomet/asm094 [18] Xue, L.G. and Zhu, L.X. (2010) Empirical Likelihood in Nonparametric and Semiparametric Models. Science Press, Beijing. [19] Cook, R.D. and Weisberg, S. (1982) Residuals and Influence in Regression. Chapman and Hall, New York. [20] Wei, B.C., Lu, G.B. and Shi, J.Q. (1990) Statistical Diagnostics. Publishing House of Southeast University, Nanjing.