Abstract
This article is concerned with the calibration of the empirical likelihood (EL) for high-dimensional data where the data dimension may increase as the sample size increases. We analyze the asymptotic behavior of the EL under a general multivariate model and provide weak conditions under which the best rate for the asymptotic normality of the empirical likelihood ratio (ELR) is achieved. In addition, there is usually substantial lack-of-fit when the ELR is calibrated by the usual normal in high dimensions, producing tests with type I errors much larger than nominal levels. We find that this is mainly due to the underestimation of the centralized and normalized quantities of the ELR. By examining the connection between the ELR and the classical Hotelling’s \(T\)-square statistic, we propose an effective calibration method which works much better in most situations.
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Acknowledgments
The authors would like to thank the Editor, Associate Editor and two anonymous referees for their many helpful comments that have resulted in significant improvements in the article. In particular, we are grateful to the Associate Editor for pointing out the necessity of Lemma 8. This research was supported by the NNSF of China Grants 11001138, 11071128, 11131002, 11101306, 11001083, the RFDP of China Grant 20110031110002 and the Fundamental Research Funds for the Central Universities. Zou thanks the support of the PAPD of Jiangsu Higher Education Institutions, and the National Center for Theoretical Sciences, Math Division.
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Liu, Y., Zou, C. & Wang, Z. Calibration of the empirical likelihood for high-dimensional data. Ann Inst Stat Math 65, 529–550 (2013). https://doi.org/10.1007/s10463-012-0384-7
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DOI: https://doi.org/10.1007/s10463-012-0384-7