Empirical likelihood for nonlinear models with missing responses

Abstract

In this paper, a nonlinear model with response variables missing at random is studied. In order to improve the coverage accuracy for model parameters, the empirical likelihood (EL) ratio method is considered. On the complete data, the EL statistic for the parameters and its approximation have a χ² asymptotic distribution. When the responses are reconstituted using a semi-parametric method, the empirical log-likelihood on the response variables associated with the imputed data is also asymptotically χ². The Wilks theorem for EL on the parameters, based on reconstituted data, is also satisfied. These results can be used to construct the confidence region for the model parameters and the response variables. It is shown via Monte Carlo simulations that the EL methods outperform the normal approximation-based method in terms of coverage probability for the unknown parameter, including on the reconstituted data. The advantages of the proposed method are exemplified on real data.

Keywords:

• empirical likelihood
• response missing at random
• random nonlinear model
• semi-parametric estimation

AMS Subject Classifications:

• 62G15
• 62F10
• 62J02
• 62G20
• 62F12

Acknowledgements

The author is grateful to an associate editor and a referee whose comments and suggestions have contributed to the improvement of the paper.