In this paper, we consider the statistical inferences for varying coefficient partially nonlinear model with missing responses. Firstly, we employ the profile nonlinear least squares estimation based on the weighted imputation method to estimate the unknown parameter and the nonparametric function, meanwhile the asymptotic normality of the resulting estimators is proved. Secondly, we consider empirical likelihood inferences based on the weighted imputation method for the unknown parameter and nonparametric function, and propose an empirical log-likelihood ratio function for the unknown parameter vector in the nonlinear function and a residual-adjusted empirical log-likelihood ratio function for the nonparametric component, meanwhile construct relevant confidence regions. Thirdly, the response mean estimation is also studied. In addition, simulation studies are conducted to examine the finite sample performance of our methods, and the empirical likelihood approach based on the weighted imputation method (IEL) is further applied to a real data example.
Citation: Liqi Xia, Xiuli Wang, Peixin Zhao, Yunquan Song. Empirical likelihood for varying coefficient partially nonlinear model with missing responses[J]. AIMS Mathematics, 2021, 6(7): 7125-7152. doi: 10.3934/math.2021418
In this paper, we consider the statistical inferences for varying coefficient partially nonlinear model with missing responses. Firstly, we employ the profile nonlinear least squares estimation based on the weighted imputation method to estimate the unknown parameter and the nonparametric function, meanwhile the asymptotic normality of the resulting estimators is proved. Secondly, we consider empirical likelihood inferences based on the weighted imputation method for the unknown parameter and nonparametric function, and propose an empirical log-likelihood ratio function for the unknown parameter vector in the nonlinear function and a residual-adjusted empirical log-likelihood ratio function for the nonparametric component, meanwhile construct relevant confidence regions. Thirdly, the response mean estimation is also studied. In addition, simulation studies are conducted to examine the finite sample performance of our methods, and the empirical likelihood approach based on the weighted imputation method (IEL) is further applied to a real data example.
[1] | T. Z. Li, C. L. Mei, Estimation and inference for varyig coefficient partially nonlinear models, J. Stat. Plann Inference, 143 (2013), 2023–2037. doi: 10.1016/j.jspi.2013.05.011 |
[2] | J. Yang, H. Yang, Smooth-threshold estimating equations for varying coefficient partially nonlinear models based on orthogonality-projection method, J. Comput. Appl. Math., 302 (2016), 24–37. doi: 10.1016/j.cam.2016.01.038 |
[3] | Y. Y. Qian, Z. S. Huang, Statistical inference for a varying-coefficient partially nonlinear model with measurement errors, Stat. Methodol., 32 (2016), 122–130. doi: 10.1016/j.stamet.2016.05.004 |
[4] | X. S. Zhou, P. X. Zhao, X. L. Wang, Empirical likelihood inferences for varying coefficient partially nonlinear models, J. Appl. Stat., 44 (2017), 474–492. doi: 10.1080/02664763.2016.1177496 |
[5] | Y. L. Jiang, Q. H. Ji, B. J. Xie, Robust estimation for the varying coefficient partially nonlinear models, J. Comput. Appl. Math., 326 (2017), 31–43. doi: 10.1016/j.cam.2017.04.028 |
[6] | Y. T. Xiao, Z. S. Chen, Bias-corrected estimations in varying-coefficient partially nonlinear models with measurement error in the nonparametric part, J. Appl. Stat., 45 (2018), 586–603. doi: 10.1080/02664763.2017.1288201 |
[7] | X. L. Wang, P. X. Zhao, H. Y. Du, Statistical inferences for varying coefficient partially nonlinear model with missing covariates, Commun. Stat. Theory Methods, (2019). |
[8] | R. J. A. Little, D. B. Rubin, Statistical analysis with missing data, 2 Eds., New York: Wiley, 2002. |
[9] | P. E. Cheng, Nonparametric estimation of mean functional with data missing at random, J. Am. Stat. Assoc., 89 (1994), 81–87. doi: 10.1080/01621459.1994.10476448 |
[10] | Q. H. Wang, J. N. K. Rao, Empirical likelihood for linear regression models under imputation for missing responses, Can. J. Stat., 29 (2001), 596–608. |
[11] | Q. H. Wang, J. N. K. Rao, Empirical likelihood-based inference under imputation for missing response data, Ann. Stat., 30 (2002), 896–924. |
[12] | Q. H. Wang, O. Linton, W. Härdle, Semiparametric regression analysis with missing response at random, J. Am. Stat. Assoc., 99 (2004), 334–345. doi: 10.1198/016214504000000449 |
[13] | L. G. Xue, Empirical likelihood for linear models with missing responses, J. Multivar. Anal., 100 (2009), 1353–1366. doi: 10.1016/j.jmva.2008.12.009 |
[14] | L. G. Xue, Empirical likelihood confidence intervals for response mean with data missing at Random, Scand. J. Stat., 36 (2009), 671–685. doi: 10.1111/j.1467-9469.2009.00651.x |
[15] | X. L. Wang, F. Chen, L. Lin, Empirical likelihood inference for estimating equation with missing data, Sci. China Math., 56 (2013), 1233–1245. doi: 10.1007/s11425-012-4504-x |
[16] | A. Owen, Empirical likelihood ratio confidence intervals for a single functional, Biometrika, 75 (1988), 237–249. doi: 10.1093/biomet/75.2.237 |
[17] | A. Owen, Empirical likelihood ratio confidence regions, Ann. Stat., 18 (1990), 90–120. |
[18] | P. P. Chen, S. Y. Feng, L. G. Xue, Statistical inference for semiparametric varying coefficient partially linear model with missing data (in Chinese), Acta Math. Sci., 35 (2015), 345–358. |
[19] | Y. P. Mack, B. W. Silverman, Weak and strong uniform consistency of kernel regression estimates, Z. Wahrsch. Verw. Gebiete., 61 (1982), 405–415. doi: 10.1007/BF00539840 |
[20] | J. Shi, T. S. Lau, Empirical Likelihood for partially linear models, Multivar. Anal., 72 (2000), 132–148. doi: 10.1006/jmva.1999.1866 |
[21] | X. L. Wang, Y. Q. Song, L. Lin, Handling estimating equation with nonignorably missing data based on SIR algorithm, J. Comput. Appl. Math., 326 (2017), 62–70. doi: 10.1016/j.cam.2017.05.016 |
[22] | X. L. Wang, Y. Q. Song, S. X. Zhang, An efficient estimation for the parameter in additive partially linear models with missing covariates, J. Korean Stat. Soc., 49 (2020), 779–801. doi: 10.1007/s42952-019-00036-6 |
[23] | H. M. Sun, Y. H. Luan, J. M. Jiang, A new classified mixed model predictor, J. Stat. Plann. Inference, 207 (2020), 45–54. doi: 10.1016/j.jspi.2019.11.001 |
[24] | Y. Fang, G. Cheng, Z. F. Qu, Optimal reinsurance for both an insurer and a reinsurer under general premium principles, AIMS Math., 5 (2020), 3231–3255. doi: 10.3934/math.2020208 |
[25] | Y. Fang, X. Wang, H. L. Liu, T. Li, Pareto-optimal reinsurance for both the insurer and the reinsurer with general premium principles, Commun. Stat. Theory Methods, 48 (2019), 6134–6154. doi: 10.1080/03610926.2018.1528364 |