The Annals of Statistics

Adjusted empirical likelihood with high-order precision

Yukun Liu and Jiahua Chen

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Empirical likelihood is a popular nonparametric or semi-parametric statistical method with many nice statistical properties. Yet when the sample size is small, or the dimension of the accompanying estimating function is high, the application of the empirical likelihood method can be hindered by low precision of the chi-square approximation and by nonexistence of solutions to the estimating equations. In this paper, we show that the adjusted empirical likelihood is effective at addressing both problems. With a specific level of adjustment, the adjusted empirical likelihood achieves the high-order precision of the Bartlett correction, in addition to the advantage of a guaranteed solution to the estimating equations. Simulation results indicate that the confidence regions constructed by the adjusted empirical likelihood have coverage probabilities comparable to or substantially more accurate than the original empirical likelihood enhanced by the Bartlett correction.

Article information

Ann. Statist., Volume 38, Number 3 (2010), 1341-1362.

First available in Project Euclid: 8 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties
Secondary: 62E20: Asymptotic distribution theory

Bartlett correction confidence region Edgeworth expansion estimating function generalized moment method


Liu, Yukun; Chen, Jiahua. Adjusted empirical likelihood with high-order precision. Ann. Statist. 38 (2010), no. 3, 1341--1362. doi:10.1214/09-AOS750.

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