Annals of the Institute of Statistical Mathematics

, Volume 65, Issue 3, pp 529–550

Calibration of the empirical likelihood for high-dimensional data

Authors

  • Yukun Liu
    • School of Statistics and FinanceEast China Normal University
    • School of Mathematical SciencesNankai University
  • Zhaojun Wang
    • School of Mathematical SciencesNankai University
Article

DOI: 10.1007/s10463-012-0384-7

Cite this article as:
Liu, Y., Zou, C. & Wang, Z. Ann Inst Stat Math (2013) 65: 529. doi:10.1007/s10463-012-0384-7

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.

Keywords

Asymptotic normality Coverage accuracy High-dimensional data Hotelling’s \(T\)-square statistic

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2012