Generalized empirical likelihood inference in partially linear model for longitudinal data with missing response variables and error-prone covariates

ABSTRACT

In this article, we consider statistical inference for longitudinal partial linear models when the response variable is sometimes missing with missingness probability depending on the covariate that is measured with error. A generalized empirical likelihood (GEL) method is proposed by combining correction attenuation and quadratic inference functions. The method that takes into consideration the correlation within groups is used to estimate the regression coefficients. Furthermore, residual-adjusted empirical likelihood (EL) is employed for estimating the baseline function so that undersmoothing is avoided. The empirical log-likelihood ratios are proven to be asymptotically Chi-squared, and the corresponding confidence regions for the parameters of interest are then constructed. Compared with methods based on NAs, the GEL does not require consistent estimators for the asymptotic variance and bias. The numerical study is conducted to compare the performance of the EL and the normal approximation-based method, and a real example is analysed.

KEYWORDS:

• Confidence region
• Generalized empirical likelihood
• Longitudinal data
• Measurement error
• Missing data
• Partially linear model

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G05
• 62G20

Acknowledgments

The authors would like to thank the referees and editors for their careful reading and helpful comments.

Funding

The second author is supported by the National Natural Science Foundation of China (11571025) and the Key Program of National Natural Science Foundation of China (11331011). The third author is supported by the National Natural Science Foundation of China (11526188), the Zhejiang Provincial Natural Science Foundation of China (NO.LQ15A010008), Startup Foundation for Talents in Zhejiang Agriculture and Forestry University (NO.2014FR085), and Statistical research project of Zhejiang Province.