Empirical likelihood inference for partial functional linear model with missing responses

ABSTRACT

In this paper, we consider the empirical likelihood inferences of the partial functional linear model with missing responses. Two empirical log-likelihood ratios of the parameters of interest are constructed, and the corresponding maximum empirical likelihood estimators of parameters are derived. Under some regularity conditions, we show that the proposed two empirical log-likelihood ratios are asymptotic standard Chi-squared. Thus, the asymptotic results can be used to construct the confidence intervals/regions for the parameters of interest. We also establish the asymptotic distribution theory of corresponding maximum empirical likelihood estimators. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals. An example of real data is also used to illustrate our proposed methods.

KEYWORDS:

• Confidence interval
• Empirical likelihood
• Functional data
• Missing data
• Partial functional linear model

MATHEMATICS SUBJECT CLASSIFICATION:

• 62M99

Acknowledgments

The authors would like to thank the Editor, the associate editor, and the referees for their helpful comments that helped to improve an earlier version of this article.

Additional information

Funding

Yuping Hu’s research was supported by the National Statistical Science Research Project of China (2014LZ45), the National Natural Science Foundation of China (11501522, 11571025). Liugen Xue’s research was supported by the National Natural Science Foundation of China (11571025, Key grant: 11331011), the Beijing Natural Science Foundation (1182002). Sanying Feng’s research was supported by the National Natural Science Foundation of China (11501522) and the Startup Research Fund of Zhengzhou University (1512315004).