Empirical Likelihood for First-order Autoregressive Error-in-variable of Models With Validation Data

Abstract

In this article, we consider the empirical likelihood for the autoregressive error-in-explanatory variable models. With the help of validation, we first develop an empirical likelihood ratio test statistic for the parameters of interest, and prove that its asymptotic distribution is that of a weighted sum of independent standard χ²₁ random variables with unknown weights. Also, we propose an adjusted empirical likelihood and prove that its asymptotic distribution is a standard χ². Furthermore, an empirical likelihood-based confidence region is given. Simulation results indicate that the proposed method works well for practical situations.

Keywords:

• Asymptotic normality
• Autoregressive model
• Confidence region
• Empirical likelihood
• Validation data.

Funding

This work is supported by National Natural Science Foundation of China (Nos. 11271155, 11371168, 11001105, 11071126, 11071269), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20110061110003), Scientific Research Fund of Jilin University (No.201100011), Jilin Province Natural Science Foundation (20130101066JC, 20130522102JH, 20101596), and Application Technology Research and Development Program Fund of Hei Longjiang Province (No. GC13D305).