Empirical likelihood for high-dimensional partially linear model with martingale difference errors

ABSTRACT

In this paper, we focus on the empirical likelihood (EL) inference for high-dimensional partially linear model with martingale difference errors. An empirical log-likelihood ratio statistic of unknown parameter is constructed and is shown to have asymptotically normality distribution under some suitable conditions. This result is different from those derived before. Furthermore, an empirical log-likelihood ratio for a linear combination of unknown parameter is also proposed and its asymptotic distribution is chi-squared. Based on these results, the confidence regions both for unknown parameter and a linear combination of parameter can be obtained. A simulation study is carried out to show that our proposed approach performs better than normal approximation-based method.

KEYWORDS:

• Confidence region
• empirical likelihood
• high-dimensional data
• martingale difference
• partially linear model

Acknowledgments

The authors are grateful to the referees for helpful comments and constructive suggestions.

Funding

This research was supported by the National Natural Science Foundation of China (11401006), the National Statistical Science Research Program of China [grant number 2015LY55], Project of Humanities and Social Science Foundation of Ministry of Education of China [grant number 15YJC910006], Project of National Bureau of Statistics of China [grant number 2016LZ05], Scientific research project of education department of Zhejiang Province [grant number Y201534425], China Postdoctoral Science Foundation [grant number 2017M611083], Visiting Program for Young Scholar in Universities [grant number gxfx2017042] and First Class Discipline of Zhejiang - A (Zhejiang Gongshang University - Statistics).