Empirical likelihood for varying coefficient partially nonlinear model with missing responses

Liqi Xia; Xiuli Wang; Peixin Zhao; Yunquan Song; Liqi Xia; Xiuli Wang; Peixin Zhao; Yunquan Song

doi:10.3934/math.2021418

AIMS Mathematics

2021, Volume 6, Issue 7: 7125-7152. doi: 10.3934/math.2021418

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Research article

Empirical likelihood for varying coefficient partially nonlinear model with missing responses

1.
School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
2.
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
3.
College of Science, China University of Petroleum, Qingdao 266580, China

Received: 20 January 2021 Accepted: 08 April 2021 Published: 28 April 2021
MSC : 62G05, 62G20

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

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Abstract

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.

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